singularities of a certain class of toric varieties1576/fulltext.pdf · singularities of a certain...

Singularities of A Certain Class of Toric

Varieties

A dissertation presented

by

Himadri Mukherjee

to

The Department of Mathematics

In partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Mathematics

Northeastern University

Boston, Massachussetts

April, 2008

1

Singularities of A Certain Class of Toric Varieties

by

Himadri Mukherjee

ABSTRACT OF DISSERTATION

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Mathematics

in the Graduate School of Arts and Sciences of

Northeastern University, April 2008

2

Abstract

Hibi considered the class of algebras k[L] = k[xα|α ∈ L] with straightening laws

associated to a finite distributive lattice L in his paper [2]. In that paper he

proves that these algebras are normal and integral domains. This result along

with the work of Sturmfels and Eisenbud [7] on binomial prime ideals implies

that the affine varieties associated to the algebra k[L] are normal toric varieties.

In the present work we will consider the toric variety X(L) = spec(k[L]), we will

give the combinatorial description of the cone σ associated to it. The final result

will be to give a standard monomial basis for the tangent cone Txτ where xτ is a

singular point associated to a torus orbit Oτ for the action of the torus T , where

τ is a face of the cone σ

3

ACKNOWLEDGMENTS

I am deeply grateful to my advisor Venkatramani Lakshmibai

for her lessons, support and patience throughout this time.

Also I would like to thank the Department of Mathematics of

Northeastern University for financial support.

4

To My Family,

baba, ma, eban EbK ipyalŇek,

jŇbn mrenr sŇmana qaĹaey

bńzu eH Aamar

reyqo daĎĹaey

5

Contents

1 Preface 8

2 Introduction 9

2.1 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Main Result 1. . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Main Result 2. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Sketch of the proof of the main result 1: . . . . . . . . . . . . . . 12

2.3 Sketch of proof of main result 2. . . . . . . . . . . . . . . . . . . . 12

2.4 Generalities on toric varieties . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Resume of combinatorics of affine toric varieties . . . . . . 15

2.4.3 Toric ideals (cf. [17]) . . . . . . . . . . . . . . . . . . . . . 17

2.4.11 Varieties defined by binomials . . . . . . . . . . . . . . . . 20

2.4.15 Orbit decomposition in affine toric varieties . . . . . . . . 22

3 Distributive lattices and Toric Varieties 24

3.1 Generalities on finite distributive lattices . . . . . . . . . . . . . . 24

3.1.10 Generalities on distributive lattices . . . . . . . . . . . . . 26

3.2 The variety X(L) . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6

3.2.5 Some more lattice theory . . . . . . . . . . . . . . . . . . . 33

4 A Basis for Cotangent Space 38

4.1 Cone and dual cone of XL: . . . . . . . . . . . . . . . . . . . . . . 38

4.1.11 Analysis of faces of σ . . . . . . . . . . . . . . . . . . . . . 42

4.2 Generation of the cotangent space at Pτ . . . . . . . . . . . . . . 45

4.3 A basis for the cotangent space at Pτ . . . . . . . . . . . . . . . . 52

4.3.1 Determination of the degree one part of J(τ): . . . . . . . 53

5 Standard Basis for Tangent Cone 56

5.1 Analysis of diamond relations: . . . . . . . . . . . . . . . . . . . . 56

5.2 Tangent cone at Pτ . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.2 Determination of I(τ): . . . . . . . . . . . . . . . . . . . . 74

7

CHAPTER 1. PREFACE 8

Chapter 1

Preface

Hibi studies a class of algebras with straightening laws in [2], which are defined

by binomial relations as a quotient of full polynomial algebras. These algebras

come up as degenerations of Schubert varieties. My work deals with constructing

a standard basis for the tangent cone to these toric varieties at a singular point.

Chapter 2

Introduction

Let K denote the base field which we assume to be algebraically closed of arbitrary

characteristic. Given a distributive lattice L, let XL denote the Hibi variety -

the affine variety in A#L whose vanishing ideal is generated by the binomials

XτXϕ − Xτ∨ϕXτ∧ϕ in the polynomial algebra K[Xα, α ∈ L]; here, τ ∨ ϕ (resp.

τ ∧ ϕ) denotes the join - the smallest element of L greater than both τ, ϕ (resp.

the meet - the largest element of L smaller than both τ, ϕ). In the sequel, we shall

refer to the quadruple (τ, ϕ, τ ∨ ϕ, τ ∧ ϕ) ((τ, ϕ) being a skew pair) as a diamond.

These varieties were extensively studied by Hibi in [13] where Hibi proves that

XL is a normal variety. On the other hand, Eisenbud-Sturmfels show in [7] that

a binomial prime ideal is toric (here, “toric ideal” is in the sense of [17]). Thus

one obtains that XL is a normal toric variety. We refer to such an XL as a Hibi

toric variety.

For L being the Bruhat poset of Schubert varieties in a minuscule G/P , it is

shown in [9] that XL flatly deforms to G/P (the cone over G/P ), i.e., there exists

a flat family over A1 with G/P as the generic fiber and XL as the special fiber.

More generally for a Schubert variety X(w) in a minuscule G/P , it is shown in

9

10

[9] that XLw flatly deforms to X(w) (here, Lw is the Bruhat poset of Schubert

subvarieties of X(w)). In a subsequent paper (cf. [11]), the authors of loc.cit.,

studied the singularities of XL,L being the Bruhat poset of Schubert varieties

in the Grassmannian; further, in loc.cit., the authors gave a conjecture giving a

necessary and sufficient condition for a point on XL to be smooth, and proved in

loc.cit., the sufficiency part of the conjecture. Subsequently, the necessary part of

the conjecture was proved in [4] by Batyrev et al. The toric varieties XL, L being

the Bruhat poset of Schubert varieties in the Grassmannian play an important

role in the area of mirror symmetry; for more details, see [3, 4]. We refer to such

an XL as a Grassmann-Hibi toric variety.

In this thesis, we first determine an explicit description (cf. §4.1) of the cone σ

associated to a Hibi toric variety and the dual cone σ∨ (note that if Sσ denotes

the semigroup of integral points in σ∨, then K[XL], the K-algebra of regular

functions on XL can be identified with the semigroup algebra K[Sσ]). Using this

we determine the cotangent space at any point on XL very explicitly as described

below.

For each face τ of σ, there exists a distinguished point Pτ in the torus orbit Oτ

corresponding to τ ; namely, identifying a (closed) point in XL with a semigroup

map Sσ → K∗∪{0}, Pτ corresponds to the semigroup map fτ which sends u ∈ Sσ

to 1 or 0 according as u is in τ⊥ or not. We have that fτ induces an algebra map

ψτ : K[XL] → K whose kernel is precisely MPτ , the maximal ideal corresponding

to Pτ . For α ∈ L, we shall denote by Pτ (α), the co-ordinate of Pτ (considered as

a point of A#L) corresponding to α. Set

Dτ = {α ∈ L |Pτ (α) 6= 0}

It turns out that Dτ is an embedded sublattice of L; further, Dτ determines the

2.1. Main Results. 11

local behavior at Pτ . To make this more precise, identifying K[XL] as the quotient

of the polynomial algebra K[Xα, α ∈ L] by the ideal generated by the binomials

XτXϕ −Xτ∨ϕXτ∧ϕ, let xα denote the image of Xα in K[XL]. For α ∈ L, set

Fα =

xα, if α 6∈ Dτ

1− xα, if α ∈ Dτ

Then we have that MPτ is generated by {Fα, α ∈ L}. Fix a maximal chain Γ in

Dτ . Let Λτ (Γ) denote the sublattice of L consisting of all maximal chains of L

containing Γ. Let Eτ denote the set of all α ∈ L such that there exists a β ∈ Dτ

such that (α, β) is a diagonal of a diamond whose other diagonal is contained in

L \ Dτ . Let Yτ (Γ) = Λτ (Γ) \ Eτ . For F ∈ MPτ , let F denote the class of F

in MPτ /M2Pτ

. We define an equivalence relation (cf. §4.2) on L \ Dτ in such a

way that for two elements θ, θ′ in the same equivalence class, we have F θ = F θ′ .

Denoting by Gτ (Γ) the set of equivalence classes [θ], where θ ∈ Yτ (Γ) \ {Γ∪Eτ}and F [θ] is non-zero in MPτ /M2

Pτ, we have

2.1 Main Results.

2.1.1 Main Result 1.

Theorem 1: (cf. Theorem 4.3.12) {F [θ], [θ] ∈ Gτ (Γ)}∪{F γ, γ ∈ Γ} is a basis for

the cotangent space MPτ /M2Pτ

.

Using the above result, we determine Sing XL (cf. Theorem 4.3.15):

Let SL = {τ < σ |#Gτ (Γ) + #Γ > #L}.

Theorem 2: (cf. Theorem 4.3.15) Sing XL = ∪τ∈SL

X(Dτ ).

(Here, X(Dτ ) denotes the Hibi variety associated to the distributive lattice Dτ .)

2.2. Sketch of the proof of the main result 1: 12

2.1.2 Main Result 2.

We first determine the vanishing ideal J(τ) of the tangent cone at a singular point

Pτ on the Hibi toric variety XL (here, Pτ is the distinguished point corresponding

to a face τ of the convex polyhedral cone associate to XL). We then introduce

a monomial order > (on the polynomial algebra containing J(τ)), and determine

in J(τ), the initial ideal of J(τ). Using MaCaulay’s Theorem [6], we obtain a

“standard monomial basis” for TCPτ XL (cf. Theorem 5.2.8).

2.2 Sketch of the proof of the main result 1:

Using our explicit description of the generators for σ, σ∨ (cf. §4.1), we first

determine explicitly the embedded sublattice associated to a face τ of σ. We

then analyze the local expression around Pτ for any f ∈ I(XL), the vanishing

ideal of XL, and show the generation of the degree one part of gr (RL,MPτ )

by {F θ, θ ∈ Yτ (Γ)} (here, RL = K[XL], and MPτ is the maximal ideal in RL

corresponding to Pτ ). We then define the equivalence relation on L \ Dτ . The

linear independence of {F [θ] [θ] ∈ Gτ (Γ)} ∪ {F γ, γ ∈ Γ} in MPτ /M2Pτ

is proved

using the defining equations of XL (as a closed subvariety of A#L), thus proving

Theorem 1. Theorem 2 is then deduced from Theorem 1.

2.3 Sketch of proof of main result 2.

Let σ be the convex polyhedral cone associate to the toric variety XL. Let Sσ be

the semigroup of integral points in σ∨ (the cone dual to σ), then K[XL], the K-

algebra of regular functions on XL can be identified with the semigroup algebra

K[Sσ]). For each face τ of σ, there exists a distinguished point Pτ in the torus

2.3. Sketch of proof of main result 2. 13

orbit Oτ corresponding to τ ; namely, identifying a (closed) point in XL with a

semigroup map Sσ → K∗ ∪ {0}, Pτ corresponds to the semigroup map fτ which

sends u ∈ Sσ to 1 or 0 according as u is in τ⊥ or not. We have that fτ induces an

algebra map ψτ : K[XL] → K whose kernel is precisely MPτ , the maximal ideal

corresponding to Pτ . For α ∈ L, we shall denote by Pτ (α), the co-ordinate of Pτ

(considered as a point of A#L) corresponding to α. Set

Dτ = {α ∈ L |Pτ (α) 6= 0}

It turns out that Dτ is an embedded sublattice of L; further, Dτ determines the

local behavior at Pτ . To make this more precise, identifying K[XL] as the quotient

of the polynomial algebra K[Xα, α ∈ L] by the ideal generated by the binomials

XτXϕ −Xτ∨ϕXτ∧ϕ, let xα denote the image of Xα in K[XL]. For α ∈ L, set

Fα =

xα, if α 6∈ Dτ


Then we have that MPτ is generated by {Fα, α ∈ L}.

Fix a maximal chain Γ in Dτ . Let Λτ (Γ) denote the sublattice of L consisting of

all maximal chains of L containing Γ. This sublattice plays a crucial role in our

discussion as explained below:

For F ∈ Mτ , let F denote the class of F in Mτ/M2τ . Then J(τ) gets identified

with the kernel of the surjective map

π : K[Xθ, θ ∈ L] → gr(R, Mτ ), Xθ 7→ Fθ

For r ∈ N, let π(r) be the restriction of π to the degree r part of the polynomial

algebra K[Xθ, θ ∈ L]. Then one of our main results is that for a diamond D

with (τ, ϕ), the diamond relation fD := XτXϕ − Xτ∨ϕXτ∧ϕ a diamond relation

2.4. Generalities on toric varieties 14

is in the ideal generated by ker π(1) and the diamond relations arising from

diamonds in Λτ (Γ). This result together with the description of the cotangent

space as determined in [15] reduces our discussion to an analysis of the diamond

relations arising from diamonds in the distributive lattice Λτ (Γ); notice one

obvious simplification, namely, by going to Λτ (Γ), we have “shrunk” Dτ to just

one chain (namely Γ) in Dτ . The precise analysis of the diamond relations arising

from diamonds in Λτ (Γ) enables us to determine J(τ) explicitly. The details will

be discussed in chapter 5

The sections are organized as follows: In §2.4, we recall generalities on toric

varieties. In §3.1, we recall some basic results on distributive lattices. In §3.2, we

introduce the Hibi variety X(L), and recall some basic results on X(L). In §4.1,

we determine generators for the cone σ (and the dual cone σ∧); further, for a face

τ of σ, we introduce Dτ and derive some properties of Dτ . In §4.2, we present our

main results on the tangent and cotangent spaces at the distinguished point Pτ

in the orbit corresponding to τ . In §5.1 we study the diamond relations in view

of the local descriptions. tangent cone at point Pτ in §5.2, we describe the ideal

generators and the standard basis for the tangent cone in §5.2.2.

2.4 Generalities on toric varieties

Since our main object of study is a certain affine toric variety, we recall in this

section some basic definitions on affine toric varieties. Let T = (K∗)m be an

m-dimensional torus.

Definition 2.4.1. (cf. [8], [14]) An equivariant affine embedding of a torus T is

an affine variety X ⊆ Al containing T as an open subset and equipped with a

T -action T ×X → X extending the action T ×T → T given by multiplication of


the group structure of T . If in addition X is normal, then X is called an affine

toric variety.

2.4.2 Resume of combinatorics of affine toric varieties

Let M be the character group of T . Let X be a toric variety with a torus action

by T ; let R = K[X]. We note the following:

• M-gradation: We have an action of T on R, and hence writing R as a

sum of T weight spaces, we obtain a M -grading for R: R = ⊕{χ∈M}

Rχ, where

Rχ = {f ∈ R|tf = χ(t)f, ∀t ∈ T}.

• The semi group S: Let S = {χ ∈ M |Rχ 6= 0}. Then via the multiplication

in R, S acquires a semi group structure. Thus S is a semi subgroup of M , and R

gets identified with the semi group algebra K[S].

• Finite generation of S: In view of the fact that R is a finitely generated

K-algebra, we obtain that S is a finitely generated semi subgroup of M.

• Generation of M by S: The fact that T and X have the same function field

(since T is a dense open subset of X) implies that M is generated by S.

• Saturation of S: The normality of R (being identified as the semi group

algebra K[S]) implies that S is saturated (recall that a semi subgroup A in M

is saturated, if for χ ∈ M , rχ ∈ A =⇒ χ ∈ A (where r is an integer > 1;

equivalently, A is precisely the lattice points in the cone generated by A, i.e.,

A = θ ∩ A where θ =∑

aixi, ai ∈ R+, xi ∈ A)).

Thus given an affine toric variety X (with torus action by T ), X determines a

finitely generated, saturated semi subgroup S of M which generates M in such a

way that the semi group algebra is the ring of regular functions on X.

Let us also observe that starting with a finitely generated, saturated semi


subgroup S of M which generates M , SpecK[S] is an affine toric variety (with

torus action by T ).

This sets up a bijection between {affine toric varieties with torus action by T}and {finitely generated, saturated semi subgroups of M which generate M}.

See [14] for details.

Let θ :=∑

aixi, ai ∈ R+, xi ∈ be the cone generated by S. Then θ is a

rational convex polyhedral cone (i.e., a cone with a (finite) set of lattice points

as generators), and θ is not contained in any hyperplane in MR := M ⊗ R. Let

σ := θ∨ = {v ∈ M∗R|v(f) ≥ 0,∀f ∈ θ}

(here, M∗R is the linear dual of MR) Then σ is a strongly convex rational convex

polyhedral cone (one may take this to be the definition of a strongly convex

rational convex polyhedral cone, namely, a rational convex polyhedral cone τ is

strongly convex if τ∨ is not contained in any hyperplane; equivalently, τ does not

contain any non-zero linear subspace.)

Thus an affine toric variety X (with a torus action by T ) determines a rational

convex polyhedral cone σ in such a way that K[X] is the semigroup algebra K[Sσ],

where Sσ is the semi subgroup in M consisting of the set of lattice points in σ∨.

Conversely, starting with a strongly convex rational convex polyhedral cone σ,

Sσ := σ∨ ∩ M is a finitely generated, saturated semi subgroup of M which

generates M ; and hence determines an affine toric variety X (with a torus action

by T ), namely, X = Spec K[Sσ].


2.4.3 Toric ideals (cf. [17])

Let A = {χ1, . . . , χl} be a subset of Zd. Consider the map

πA : Zl+ → Zd, u = (u1, . . . , ul) 7→ u1χ1 + · · ·+ ulχl.

Let K[x] := K[x1, . . . , xl], K[t±1] := K[t1, . . . , td, t−11 , . . . , t−1

d ].

The map πA induces a homomorphism of semigroup algebras

πA : K[x] → K[t±1], xi 7→ tχi .

Definition 2.4.4. (cf. [17]) The kernel of πA is denoted by IA and called the

toric ideal associated to A.

Note that a toric ideal is prime. We shall now show that we have a bijection

between the class of toric ideals and the class of equivariant affine toroidal

embeddings.

Consider the action of T (= (K∗)d) on Al given by

t(a1, · · · , αl) = (tχ1a1, · · · , tχlal)

Then V(IA), the affine variety of the zeroes in K l of IA, is simply the Zariski

closure of the T -orbit through (1, 1, . . . , 1). Let MA be the Z-span of A (inside

X(T ) = Zd), and dA the rank of MA. Let SA be the sub semigroup of Zd generated

by A. Let TA be the dA-dimensional torus with MA as the character group. Then

we have

Proposition 2.4.5. V(IA) is an equivariant affine embedding of TA (of dimension

dA); further, K[V(IA)] is the semigroup algebra K[SA].

Remark 2.4.6. In the above definition, we do not require V(IA) to be normal.

Note that V(IA) is normal if and only if SA is saturated. A variety of the form


V(IA), is also sometimes called an affine toric variety. But in this paper, by an

affine variety, we shall mean a normal toric variety as defined in Definition 2.4.1.

Remark 2.4.7. Consider the action of T (= (K∗)d) on K l given by tei = tχiei

(here, ei, 1 ≤ i ≤ l are the standard basis vectors of K l). Then V(IA), the

affine variety of the zeroes in K l of IA is simply the Zariski closure of the T -orbit

through (1, 1, . . . , 1). In particular, V(IA) is an equivariant affine embedding of

T = (K∗)d.

The vanishing ideal of X

Let us return to the affine toric variety X ↪→ Al with a torus action by T (= (K)∗)d

as in Definition 2.4.1. Let Tl = (K∗)l (an l-dimensional torus). Consider T as

a subtorus of Tl. The natural action of Tl on Al induces an action of T on Al,

and the embedding X ↪→ Al is T -equivariant. Hence if we denote the images

of {x1, · · · , xl} in K[X] by {y1, · · · , yl}, then {y1, · · · , yl} are T -weight vectors

with weights {χ1, · · · , χl}, say. Then {χ1, · · · , χl} is a set of generators for the

semi group Sσ (notation being as in §2.4.2, namely, K[X] equals the semi group

algebra K[Sσ]). Denoting, A := {χ1, · · · , χl}, we have that IA of Definition 2.4.4

is precisely the vanishing ideal of X. Thus we obtain

Proposition 2.4.8. I(X), the vanishing ideal of X (for the embedding X ↪→ Al)

is a toric ideal.

Remark 2.4.9. Note that we also have the saturation property for A (since Sσ

is a saturated sub semigroup of X(T ))

Recall the following (see [17]).


Proposition 2.4.10. The toric ideal IA is spanned as a K-vector space by the

set of binomials

{xu − xv | u,v ∈ Zn+ with πA(u) = πA(v)}. (*)

(Here, a binomial is a polynomial with at most two terms.)

An example

Let us fix the integers n1, . . . , nd > 1, and let n = Πdi=1ni, m =

∑di=1 ni. Let

el1, . . . , e

lnl

be the unit vectors in Znl , for 1 ≤ l ≤ d. For 1 ≤ ξ1 ≤ n1, . . . , 1 ≤ξd ≤ nd, define

aξ1...ξd= e1

ξ1+ · · ·+ ed

ξd∈ Zn1 ⊕ · · · ⊕ Znd

and let

An1,...,nd= {aξ1...ξd

| 1 ≤ ξ1 ≤ n1, . . . , 1 ≤ ξd ≤ nd}.

The corresponding map

πA : Zn1·····nd+ → Zn1+···+nd

is defined as follows: for 1 ≤ l ≤ d and 1 ≤ il ≤ nl fixed, the (n1+· · ·+nl−1+il)-th

coordinate of πA(u) is given by∑

uξ1...ξl−1ξlξl+1...ξd, the sum being taken over the

elements (ξ1, . . . , ξl−1, ξl, ξl+1, . . . , ξd) of C(n1, . . . , nd) with ξl = il. We call this

subset the l-th slice of C(n1, . . . , nd) defined by il, and denote it by {ξl = il}. The

components (or entries) of an element u ∈ Zn1·····nd are indexed by the elements

(i1, . . . , id) of C(n1, . . . , nd). If (j1, . . . , jd) ∈ {ξl = il}, sometimes we also say that

uj1...jditself belongs to the slice {ξl = il}.

The map πA induces the map

πA : k[x11...1, . . . , xξ1ξ2...ξd, . . . , xn1n2...nd

] → k[t11, . . . , t1n1 , . . . , td1, . . . , tdnd]


given by

xξ1...ξd7→ t1ξ1 . . . tdξd

, for 1 ≤ ξ1 ≤ n1, . . . , 1 ≤ ξd ≤ nd.

2.4.11 Varieties defined by binomials

Let l ≥ 1, and let X be an affine variety in Al, not contained in any of the

coordinate hyperplanes {xi = 0}. Further, let X be irreducible, and let its

defining prime ideal I(X) be generated by l binomials

xai11 . . . xail

l − λixbi11 . . . xbil

l , 1 ≤ i ≤ m . (∗)

Consider the natural action of the torus Tl = (K∗)l on Al,

(t1, . . . , tl) · (a1, . . . , al) = (t1a1, . . . , tlal).

Let X(Tl) = Hom(Tl,Gm) be the character group of Tl, and let εi ∈ X(Tl) be the

character

εi(t1, . . . , tl) = ti , 1 ≤ i ≤ l.

For 1 ≤ i ≤ m, let

ϕi =l∑

t=1

(ait − bit)εt .

Set T = ∩mi=1 ker ϕi, and X◦ = {(x1, . . . , xl) ∈ X | xi 6= 0 for all 1 ≤ i ≤ l}.

Proposition 2.4.12. Let notations be as above.

(1) There is a canonical action of T on X.

(2) X◦ is T -stable. Further, the action of T on X◦ is simple and transitive.

(3) T is a subtorus of Tl, and X is an equivariant affine embedding of T .

Proof 2.4.13. (1) We consider the (obvious) action of T on Al. Let (x1, . . . , xl) ∈X, t = (t1, . . . , tl) ∈ T , and (y1, . . . , yl) = t · (x1, . . . , xl) = (t1x1, . . . , tlxl). Using


the fact that (x1, . . . , xl) satisfies (∗), we obtain

yai11 . . . yail

l = tai11 . . . tail

l xai11 . . . xail

l = λitbi11 . . . tbil

l xbi11 . . . xbil

l = λiybi11 . . . ybil

l ,

for all 1 ≤ i ≤ m, i.e. (y1, . . . , yl) ∈ X. Hence t · (a1, . . . , al) ∈ X for all t ∈ T .

(2) Let x = (x1, . . . , xl) ∈ X◦, and t = (t1, . . . , tl) ∈ T . Then, clearly

t · (x1, . . . , xl) ∈ X◦. Considering x as a point in Al, the isotropy subgroup

in Tl at x is {id}. Hence the isotropy subgroup in T at x is also {id}. Thus the

action of T on X◦ is simple.

Let (x1, . . . , xl), (x′1, . . . , x′l) ∈ X◦. Set t = (t1, . . . , tl), where ti = xi/x

′i. Then,

clearly t ∈ T . Thus (x1, . . . , xl) = t · (x′1, . . . , x′l). Hence the action of T on X◦ is

simple and transitive.

(3) Now, fixing a point x ∈ X◦, we obtain from (2) that the orbit map t 7→ t · xis in fact an isomorphism of T onto X◦. Also, since X is not contained in any

of the coordinate hyperplanes, the open set Xi = {(x1, . . . , xl) ∈ X | xi 6= 0}is nonempty for all 1 ≤ i ≤ l. The irreducibility of X implies that the sets Xi,

1 ≤ i ≤ l, are open dense in X, and hence their intersection

X◦ =l⋂

i=1

Xi = {(x1, . . . , xl) ∈ X | xi 6= 0 for any i}

is an open dense set in X, and thus X◦ is irreducible. This implies that T is

irreducible (and hence connected). Thus T is a subtorus of Tl. The assertion that

X is an equivariant affine embedding of T follows from (1) and (2).

Remark 2.4.14. One can see that the ideal I(X) is a toric ideal in the sense of

Definition 2.4.4; more precisely, A = {ρi, 1 ≤ i ≤ l}, where ρi = εi

∣∣T

(here K[T ]

is identified with K[t±11 , . . . , t±1

d ], and the character group X(T ) with Zd).


2.4.15 Orbit decomposition in affine toric varieties

We shall preserve the notation of §2.4.2. As in §2.4.2, let X be an affine toric

variety with a torus action by T . Let K[X] = K[Sσ]. We shall denote X also by

Xσ.

Let us first recall the definition of faces of a convex polyhedral cone:

Definition 2.4.16. A face τ of σ is a convex polyhedral sub cone of σ of the

form τ = σ ∩ u⊥ for some u ∈ σ∨, and is denoted τ < σ.

We have that Xτ is a principal open subset of Xσ, namely,

Xτ = (Xσ)f

Each face τ determines a (closed) point Pτ in Xσ, namely, it is the point

corresponding to the maximal ideal in K[X](= K[Sσ]) given by the kernel of

eτ : K[Sσ] → K, where for u ∈ Sσ, we have

eτ (u) =

1, if u ∈ τ⊥

0, otherwise

Remark 2.4.17. As a point in Al, Pτ may be identified with the l-tuple with 1

at the i-th place if χi is in τ⊥, and 0 otherwise (here, as in §2.4.3, χi denotes the

weight of the T -weight vector yi - the class of xi in K[Xσ]).

Let Oτ denote the T -orbit in Xσ through Pτ . We have the following orbit

decomposition in Xσ:

Xσ = ∪θ≤σ

Oθ

Oτ = ∪θ≥τ

Oθ

dim τ + dimOτ = dimXσ


(here, by dimension of a cone τ , one means the vector space dimension of the

span of τ).

See [8], [14] for details.

Thus τ 7→ Oτ defines an order reversing bijection between {faces of σ} and {T -

orbit closures in Xσ}. In particular, we have the following two extreme cases:

1. If τ is the 0-face, then Pτ = (1, · · · , 1) (as a point in Al), and Oτ = T , and is

contained in Xθ,∀θ < σ. It is a dense open orbit.

2. If τ = σ, then Pτ (as a point in Al) is the l-tuple with 1 at the i-th place if χi

is in τ⊥, and 0 otherwise, and Oτ = {Pτ}, and is the unique closed orbit.

For a face τ , let us denote by Nτ the sublattice of N (the Z-dual of M) generated

by the lattice points of τ . Let N(τ) = N/Nτ , and M(τ), the Z-dual of N(τ). For

a face θ of σ such that θ contains τ as a face, set

θτ := (θ + (Nτ )R)/(Nτ )R

(here, for a lattice L, LR = L⊗R). Then the collection {θτ , σ > θ > τ} gives the

set of faces of the cone στ (⊂ N(τ)R).

Lemma 2.4.18. For a face τ < σ, Oτ gets identified with the toric variety

SpecK[Sστ ]. Further, K[Oτ ] = K[Sσ ∩ τ⊥].

Proof 2.4.19. The first assertion follows from the description of the orbit

decomposition ( and the definition of στ ). For the second assertion, we have

Sστ = σ∨τ ∩M(τ) = σ∨ ∩ (τ⊥ ∩M) = Sσ ∩ τ⊥

(note that we have, M(τ) = τ⊥ ∩M).

Chapter 3

Distributive lattices and Toric

Varieties

3.1 Generalities on finite distributive lattices

In this section we recall elementary properties of Distributive lattice.

A partially ordered set which sometime also called a poset is defined as a set p

with a transitive relation ≥. If x ≥ y, in phrase we call x is greater than y or y

is less than x. If x, y ∈ P and x ≥ y and x 6= y then we call x is strictly greater

than y, and write it as x > y. An element x ∈ P is called maximal if x is greater

than all y ∈ P . In the case when an element x ∈ P is less than all the elements

y ∈ P then we y the minimal element of the poset.

Definition 3.1.1. A finite poset P is called bounded if there exist both maximal

and minimal elements and are unique. For the rest of the text we denote 1 as the

unique maximal element of a bounded poset. and 0 for the minimal element of

the same bounded poset.

24

3.1. Generalities on finite distributive lattices 25

Example Let S be a non empty set with finite elements. Let P(S) be the set of

subsets of S, then the inclusion among the subsets define a partial order. A ≥ B

if A ⊂ B. In this poset there is a minimal element Φ and the set S considered

as a subset of itself is a maximal element which are clearly unique. This gives an

example of a bounded poset.

Definition 3.1.2. A totally ordered subset C of a finite poset P is called a chain,

and the number #C − 1 is called the length of the chain.

A chain in a bounded poset p is called a maximal chain if it contains both the

minimal and the maximal element.

Definition 3.1.3. A bounded poset P is said to be graded (or also ranked) if

all maximal chains have equal length (note that 1 and 0 belong to any maximal

chain).

Example The bounded poset in the above example is also a ranked poset; each

maximal chain has length equal to the number of elements in the set S.

Definition 3.1.4. Let P be a graded poset. The length of a maximal chain in P

is called the rank of P .

Definition 3.1.5. In a graded poset P for elements λ, µ ∈ P with λ ≥ µ, the

poset {τ ∈ P | µ ≤ τ ≤ λ} is called the interval from µ to λ, and denoted by

[µ, λ]. The interval is clearly a graded poset. We define the rank of [µ, λ] by lµ(λ);

if µ = 0, then we denote lµ(λ) by just l(λ).

Theorem 3.1.6. Let P be a graded poset, x, y ∈ P two element x ≥ y then the

interval [y, x] is a graded poset.

Proof 3.1.7. Lets call the interval I, I gets the poset structure from the poset

relation of the larger poset P , its also bounded since the element x is the maximal


element where y is the minimal element of the poset. To prove that all the

maximal chains have same length lets assume that M1,M2 are two maximal chains

with different lengths. We give an order relation ≥ on the totally ordered subsets

of the graded poset P given by the inclusion. Under this order any inductively

ordered chain must have a maximal element since the set of all totally ordered

chains in P is finite. Note that this maximal is a maximal chain of the poset P .

So we can extend any totally ordered chain of the lattice P to a maximal chain.

Remark 3.1.8. In the later part of the work, we refer to l(λ) as the level of λ in

L.

Definition 3.1.9. Let P be a graded poset, and λ, µ ∈ P , with λ ≥ µ. The

ordered pair (λ, µ) is called a cover (and we also say that λ covers µ) if lµ(λ) = 1.

3.1.10 Generalities on distributive lattices

Definition 3.1.11. A lattice is a partially ordered set (L,≤) such that, for every

pair of elements x, y ∈ L, there exist elements x ∨ y and x ∧ y, called the join,

respectively the meet of x and y, defined by:

x ∨ y ≥ x, x ∨ y ≥ y, and if z ≥ x and z ≥ y, then z ≥ x ∨ y,

x ∧ y ≤ x, x ∧ y ≤ y, and if z ≤ x and z ≤ y, then z ≤ x ∧ y.

It is easy to check that the operations ∨ and ∧ are commutative and associative.

Definition 3.1.12. An element z ∈ L is called the zero of L, denoted by 0, if

z ≤ x for all x in L. An element z ∈ L is called the one of L, denoted by 1, if

z ≥ x for all x in L.


Definition 3.1.13. Given a lattice L, a subset L′ ⊂ L is called a sublattice of L

if x, y ∈ L′ implies x ∧ y ∈ L′, x ∨ y ∈ L′.

Definition 3.1.14. Two lattices L1 and L2 are em isomorphic if there exists a

bijection ϕ : L1 → L2 such that, for all x, y ∈ L1,

ϕ(x ∨ y) = ϕ(x) ∨ ϕ(y) and ϕ(x ∧ y) = ϕ(x) ∧ ϕ(y).

Definition 3.1.15. A lattice is called distributive if the following identities hold:

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) (3.1)

x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). (3.2)

Example of a distributive lattice would be the poset of the powerset of a set S.

Where the operations ∨ and ∧ are given by union and intersection respectively.

Definition 3.1.16. An element z of a lattice L is called join-irreducible (resp.

meet irreducible ) if z = x ∨ y (respectively z = x ∧ y) implies z = x or z = y.

The set of join-irreducible (resp. meet irreducible) elements of L is denoted by

JL (resp. ML), or just by J (resp. M) if the reference to the lattice L is obvious

in the context.

Examples: 1. The lattice of all subsets of the set {1, 2, . . . , n} is denoted by

B(n), and called the Boolean algebra of rank n.

A good class of example of distributive lattice would be the chain product lattices

defined below.

2. Chain product lattice: Given an integer n ≥ 1, C(n) C(n) will denote

the chain {1 < · · · < n},C(n1, . . . , nd) and for n1, . . . , nd > 1, C(n1, . . . , nd) will

denote the chain product lattice C(n1) × · · · × C(nd) consisting of all d-tuples


(i1, . . . , id), with 1 ≤ i1 ≤ n1, . . . , 1 ≤ id ≤ nd. For (i1, . . . , id), (j1, . . . , jd) in

C(n1, . . . , nd), we define

(i1, . . . , id) ≤ (j1, . . . , jd) ⇐⇒ i1 ≤ j1, . . . , id ≤ jd .

We have

(i1, . . . , id) ∨ (j1, . . . , jd) = (max{i1, j1}, . . . , max{id, jd})

(i1, . . . , id) ∧ (j1, . . . , jd) = (min{i1, j1}, . . . , min{id, jd}).C(n1, . . . , nd) is a finite distributive lattice, and its zero and one are (1, . . . , 1),

(n1, . . . , nd) respectively.

Note that there is a total order / on C(n1, . . . , nd) extending <, namely the

lexicographic order, defined by (i1, . . . , id) / (j1, . . . , jd) if and only if there exists

l < d such that i1 = j1, . . . , il = jl, il+1 < jl+1. Also note that two elements

(i1, . . . , id) / (j1, . . . , jd) are non-comparable with respect to ≤ if and only if there

exists 1 < h ≤ d such that ih > jh.

Sometimes we denote the elements of C(n1, n2, . . . , nd) by xi1...id , with 1 ≤ i1 ≤n1, . . . , 1 ≤ id ≤ nd.

In the case x y ∈ L, where L is a lattice, when x is neither larger nor less than y,

we call x and y non comparable.

One has the following embedding of distributive lattices into sublattices of boolean

algebra. (see [1]):

Theorem 3.1.17. Any finite distributive lattice is isomorphic to a sublattice of

a Boolean algebra of finite rank, and hence, in particular, to a sublattice of a

finite chain product lattice.

We will soon concentrate on the maximal chains of a distributive lattice, and in

turn we will build up these maximal chains with covers. As a good understanding

of the covers the following easy corollary to the definition of a cover will be useful.


Corollary 3.1.18. Let (τ, λ1), (τ, λ2) be two covers in a distributive lattice L.

Then λ1 ∧ λ2 is covered by both λ1 and λ2

Definition 3.1.19. A sublattice of the form {τ, φ, τ ∨ φ, τ ∧ φ}, with τ, φ ∈ L

non-comparable is called a diamond, and is denoted by D{τ, φ, τ ∨ φ, τ ∧ φ} or

also just D(τ, φ). The pair (τ, φ) (respectively (τ ∨ φ, τ ∧ φ))) is called the skew

(respectively main) diagonal of the diamond D(τ, φ).

A diamond of the form {x∨ y, x, y, x∧ y} where x and y are non comparable and

x∨y covers both x and y, hence x and y both covers x∧y by the above corollary,

will be called a simple diamond.

Definition 3.1.20. A subset I of a poset P is called an ideal of P if for all

x, y ∈ P ,

x ∈ I and y ≤ x imply y ∈ I.

Theorem 3.1.21. (Birkhoff) Let L be a distributive lattice with 0, and P the

poset of its nonzero join-irreducible elements. Then L is isomorphic to the lattice

of finite ideals of P , by means of the lattice isomorphism

α 7→ Iα = {τ ∈ P | τ ≤ α}, α ∈ L.

Since in the later part of the work we would be required to collect the set of all

non comparable pairs in a particular distributive lattice, L we set.

Q(L) = {(τ, ϕ), τ, ϕnoncomparable}

In the sequel, Q(L) will also be denoted by just Q when there is no possible

confusion.

We prove the following lemma to give another criterion for the join irreducibles

in terms of the language of covers.


Lemma 3.1.22.

JL = {τ ∈ L | there exists at most one cover of the form (τ, λ)}.

Proof 3.1.23. The proof is clear from the observation that if τ ∈ L covers

more than one element, let us say two of them are α1, α2, then, consider α1 ∨ α2

which is larger than both α1 and α2. But since τ ≥ α1 and τ ≥ α2, we

have τ ≥ α1 ∨ α2 ≥ α1 but since τ was a cover for α1 this means that either

α1 = α1 ∨ α2 which in turn implies that α1 ≥ α2. Once again the repeated

argument considering that τ is a cover of α2 we must have α1 = α2 contrary

to our assumption that these are two different elements. On the other hand if

α1 ∨ α2 = τ then τ being the join of these two is no longer a join irreducible

contradicting our hypothesis. This ends the proof.

For α ∈ L, let Iα be the ideal corresponding to α under the isomorphism in

Theorem 3.1.21.

Lemma 3.1.24. Let (τ, λ) be a cover in L. Then Iτ equals Iλ∪{β} for some

β ∈ JL.

Proof 3.1.25. If τ ∈ JL, then λ is the unique element covered by τ ; it is clear

that in this case that Iτ = Iλ ∪ {τ} since by definition Iτ = {α, α ≤ τ} which is

same as {τ} ∪ {α ≤ λ} = Iτ ∪ {τ}.

Let then τ 6∈ JL. By the previous lemma we know that τ must cover more than

one element. Let λ′ be another element covered by τ . Let φ = λ ∧ λ′. Now

note that φ = τ since τ covers λ and we have τ ≥ φ ≤ λ which is possible only

when τ = φ. Now by induction on the level of τ , we have, Iλ = Iφ ∪ {β}, for

some β ∈ JL (note that (λ, φ), (λ′, φ) are both covers in view of Corollary 3.1.18).

Then Iλ′ ∪{β} is an ideal contained in Iτ (note that if γ in JL is such that γ < β,

then γ is in Iλ, and hence in Iτ ). Hence if θ is the element of L corresponding to

3.2. The variety X(L) 31

the ideal Iλ′ ∪ {β} (under the bijection given by Theorem 3.1.21), then we have,

λ′ < θ ≤ τ . Hence, we obtain that θ = τ (since (τ, λ′) is a cover).

Starting point of induction: Let τ be an element of least length among all θ’s

such that θ covers an element θ′. Then the above reasoning implies clearly that

τ ∈ JL, and therefore, Iτ = Iλ ∪ {τ}.

Note that as a corollary to the above lemma we get that the length of a maximal

chain is less than equal to the number of elements in the join irreducible set. In

fact we have the following proposition.

Proposition 3.1.26. Any maximal chain in (the ranked poset) L has cardinality

equal to #JL.

Proof 3.1.27. which follows from the observation I1 equals JL along with the

lemma 3.1.24

3.2 The variety X(L)

Consider the polynomial algebra K[xα, α ∈ L]; let I(L) be the ideal generated by

{xαxβ − xα∨βxα∧β, α, β ∈ L}. Then one knows (cf.[13]) that K[xα, α ∈ L] /I(L)

is a normal integral domain; in particular, we have that I(L) is a prime ideal.

Let X(L) be the affine variety of the zeroes in Kn of I(L) (where, n = #L).

Since the ideal I(L) is a binomial ideal we can say that X(L) is a affine normal

variety defined by binomials, and hence by §we know that the variety is a Toric

embedding. That is its an affine variety with a dense torus T with an action

on X(L) extending the action of T on T . We also know that since the toric

embedding X(L) is normal it must be a toric variety associated to a normal cone.


To discuss the singularity of the variety X(L) we will first calculate the dimension

of X(L). Since a toric variety X contains a dense torus T the dimension of X is

same as the dimension of the torus T . To compute the dimension of the variety

X(L) we will compute the dimension of the torus T .

Before we calculate the dimension we must define a suitable combinatorial set,

which will be essential later. Let

I = {(τ, φ, τ ∨ φ, τ ∧ φ) | (τ, φ) ∈ Q}, where

Q = {(τ, φ) | τ, φ ∈ L non-comparable}.

Let Tl = (K∗)l, π : X(Tl) → X(T ) be the canonical map, given by restriction,

and for χ ∈ X(Tl), denote π(χ) by χ. Let us fix a Z-basis {χτ | τ ∈ L} for X(Tl).

For a diamond D = (τ, φ, τ ∨ φ, τ ∧ φ) ∈ I, let χD = χτ∨φ + χτ∧φ − χτ − χφ.

Lemma 3.2.1. We have

(1) X(T ) ' X(Td)/ ker π.

(2) ker π is generated by {χD | D ∈ I}.

Proof 3.2.2. The canonical map π is, in fact, surjective, since T is a subtorus of

Td. Now (1) follows from this. The assertion (2) follows from the definition of T .

Let X(JL) be the Z-span of {χθ, θ ∈ JL}. As an immediate consequence of the

above Lemma, we have

Corollary 3.2.3. π maps X(JL) isomorphically onto its image.

Proof 3.2.4. The result follows since there cannot be a diamond D contained

completely in JL.


3.2.5 Some more lattice theory

For α ∈ L, let Iα be the ideal corresponding to α under the isomorphism in

Theorem 3.1.21. Let us define

ψα =∑

θ∈Iα

χθ.

Taking a total order on L extending the partial order, we see easily that the

transition matrix expressing ψα’s in terms of χθ’s is triangular with diagonal

entries equal to 1. Let α1 ≥ α2 ≥ . . . . . . ≥ αn denote the total order on the

lattice L extending the partial order relation on L. Further if we call δα,β the

indicator function whether xβ occurs in ψα. That is δα,β = 1 if and only if xβ is

a summand of ψα. Then we have the following matrix:

Ψ =

ψα1

ψα2

...

ψαn

= (δα,β)

Since the matrix is upper triangular with diagonal entries equal to 1 we have the

following obvious lemma.

Lemma 3.2.6. {ψα, α ∈ L} is a Z-basis for X(Tn) the torus Tn being embedded

as an open set in the affine space An.

Clearly Since ψα is a basis for the torus Tn and the torus T embedded into it, we

get the following lemma.

If we consider the surjective (restriction) map π : X(Tn) → X(T ), and denoting

π (ψα) by ψα, we obtain

Lemma 3.2.7. The set {ψα | α ∈ L} generates X(T ) as a Z-module.


Since we know that rank of T is equal to the rank of the character group X(T ).

Calculating the dimension of the kernel of the surjection map will be sufficient to

calculate the dimension of the torus T .

Lemma 3.2.8. Let α ∈ L. Then ψα is in the Z-span of {χθ, θ ≤ α, θ ∈ JL}. In

particular, {χθ, θ ≤ α, θ ∈ J} generates X(T ) as a Z-module.

Proof 3.2.9. (by induction)

Case 1: α ∈ JL (may suppose α 6= 0, since if α = 0, then the result is clear).

There exists a unique β ∈ L covered by α. We have

ψα = χα + ψβ

By induction, ψβ belongs to the Z-span of {χθ, θ ≤ β, θ ∈ JL}. The result follows

from this.

Case 2: α 6∈ JL.

Hypothesis implies that there exist α1, α2 both of which are covered by α. Then

α1, α2 are non-comparable since if possible α ≥ α1 ≥ α2 will contradict the

hypothesis that α covers α2. We have α1 ∨ α2 = α; let β = α1 ∧ α2. We have

ψα = ψα1+ ψα2

− ψβ

The result follows by induction.

Combining the above Lemma with Corollary 3.2.3, we obtain

Proposition 3.2.10. The set {χτ | τ ∈ JL} is a Z-basis for X(T ).

Now, since dim X(L) = dim T , we obtain

Theorem 3.2.11. The dimension of X(L) is equal to #JL.


Combining the above theorem with Lemma 3.1.24 and Corollary 3.1.26, we obtain

Corollary 3.2.12. 1. The dimension of XL is equal to the cardinality of the

set of elements in a maximal chain in L.

2. Fix any chain β1 < · · · < βd, d being #JL. Let γi+1 be the element

of JL corresponding to the cover (βi+1, βi), i ≥ 1; set γ1 = β1. Then

JL = {γ1, · · · , γd}

Definition 3.2.13. For a finite distributive lattice L, we call the cardinality of

JL the dimension of L, and we denote it by dim L.

Lemma 3.2.14. Let P = (Pθ)θ∈L ∈ X(L) be such that Pτ 6= 0 for any τ ∈ JL.

Then Pθ 6= 0 for any θ ∈ L.

Proof 3.2.15. (by induction) Let θ ∈ L. If θ ∈ JL, there is nothing to check.

Let then θ ∈ L \ JL.

Let θ be a minimal element of L \ JL. This implies that every τ ∈ L such that

τ < θ belongs to JL. The fact that θ ∈ L \ JL implies that there are at least two

elements θ1, θ2 of L which are covered by θ. Note that θ1, θ2 are not comparable.

We have θ1 ∨ θ2 = θ. Let µ = θ1 ∧ θ2. We have PθPµ = Pθ1Pθ2 . Now Pθ1 6= 0,

Pθ2 6= 0, since θ1, θ2 ∈ JL. Hence we obtain that Pθ 6= 0.

Let now φ be any element of L \ JL. Assume, by induction, that Pτ 6= 0 for any

τ < φ. Since φ 6∈ JL, there are at least two elements φ1, φ2 of L which are covered

by φ. We have φ1 ∨ φ2 = φ, PφPδ = Pφ1Pφ2 , where δ = φ1 ∧ φ2. Also Pφ1 6= 0,

Pφ2 6= 0 (since φ1, φ2 are both < φ). Hence we obtain Pφ 6= 0.

We would like to list all the toric subvarieties of the toric variety X(L), towards

that we define an important class of distributive sublattices.


Definition 3.2.16. A sublattice L′ of L is called an embedded sublattice of L if

τ, φ ∈ L, τ ∨ φ, τ ∧ φ ∈ L′ ⇒ τ, φ ∈ L′.

Given a sublattice L′ of L, let us consider the variety X(L′), and consider

the canonical embedding X(L′) ↪→ A(L′) ↪→ A(L) (here A(L′) = A#L′ ,

A(L) = A#L).

In the following lemma we classify all the toric subvarieties of the variety X(L).

Proposition 3.2.17. X(L′) is a subvariety of X(L) if and only if L′ is an

embedded sublattice of L.

Proof 3.2.18. Under the embedding X(L′) ↪→ A(L), X(L′) can be identified

with

{(xτ )τ∈L ∈ A(L) | xτ = 0 if τ 6∈ L′, xτxφ = xτ∨φxτ∧φ for τ, φ ∈ L′ noncomparable }.

Let η′ be the generic point of X(L′). We have X(L′) ⊂ X(L) if and only if

η′ ∈ X(L).

Assume that η′ ∈ X(L). Let τ , φ be two noncomparable elements of L such that

τ ∨φ, τ ∨φ are both in L′. We have to show that τ, φ ∈ L′. If possible, let τ 6∈ L′.

This implies η′τ = 0. Hence either η′τ∨φ = 0, or η′τ∧φ = 0, since η′ ∈ X(L). But

this is not possible (note that τ ∨ φ, τ ∧ φ are in L′, and hence η′τ∨φ and η′τ∧φ are

both nonzero).

Assume now that L′ is an embedded sublattice. We have to show that η′ ∈ X(L).

Let τ , φ be two non-comparable elements of L. The fact that L′ is a sublattice

implies that if η′τ∨φ or η′τ∧φ is zero, then either η′τ , or η′φ is zero. Also, the fact

that L′ is an embedded sublattice implies that if η′τ or η′φ is zero, then either η′τ∨φ

or η′τ∧φ is zero. Further, when τ, φ, τ ∨ φ, τ ∧ φ ∈ L′,

η′τη′φ = η′τ∨φη

′τ∧φ.


Thus η′ satisfies the defining equations of X(L), and hence η′ ∈ X(L).

Chapter 4

A Basis for Cotangent Space

4.1 Cone and dual cone of XL:

In this section we shall determine the cone and the dual cone of XL, L being

a finite distributive lattice. As in the previous sections, denote the poset of

join-irreducibles of L by JL; let d = # JL. Denote JL = {β1, · · · , βd}. Let T

be a d-dimensional torus. Identifying T with (K∗)d, let {fβ, β ∈ JL} denote the

standard Z-basis for X(T ), namely, for u ∈ T, u = (uβ, β ∈ JL), fβ(u) = uβ. Then

K[X(T )] may be identified with K[u±1β1

, · · · , u±1βd

], the ring of Laurent polynomials.

If χ ∈ X(T ), say χ =∑

aβfβ, then as an element of K[X(T )], χ will also be

denoted by∏

uaβ

β .

Denote by I(JL) the poset of ideals of JL. For A ∈ I(JL), set

fA :=∑z∈A

fz

Recall (cf. Theorem 3.1.21) the bijection L → I(JL), θ 7→ Aθ := {τ ∈ JL | τ ≤ θ}.

Lemma 4.1.1. {fAθ, θ ∈ JL} is a Z-basis for X(T ).

38

4.1. Cone and dual cone of XL: 39

Proof 4.1.2. Take a total order on JL extending the partial order. Then the

matrix expressing {fAθ, θ ∈ JL} in terms of the basis {fθ, θ ∈ JL} is easily seen

to be triangular with diagonal entries equal to 1. The result follows from this.

Let A = {fA, A ∈ I(JL)}. Then as a consequence of the above Lemma, we obtain

Corollary 4.1.3. A generates X(T ); in particular, dA = d, dA being as in

Remark 2.4.7.

For A ∈ I(JL), denote by mA the monomial:

mA :=∏τ∈A

uτ

in K[X(T )]. If α is the element of L such that Iα = A (cf. Theorem 3.1.21), then

we shall denote mA also by mα. Consider the surjective algebra map

F : K[Xα, α ∈ L] → K[mA, A ∈ I(JL)] (⊂ K[X(T )]), Xα 7→ mA, A = Iα

Then with notation as in Definition 2.4.4, we have, F = πA, ker F = IA. Further,

Proposition 2.4.7 and Corollary 4.1.3 imply the following

Proposition 4.1.4. V(IA) = SpecK[mA, A ∈ I(JL)] and is of dimension

d(= #JL) (here, V(IA) is as in Remark 2.4.7)

Let us denote V(IA) by Y .

Lemma 4.1.5. Kernel of F is generated by {XαXβ −Xα∨βXα∧β, α, β ∈ L}

Proof 4.1.6. We have (in view of Lemma 3.1.24) that if (α, λ) is a cover, then

Iα equals Iλ∪{β} for some β ∈ JL. A repeated application of this result implies

that if γ ≤ α, and m = l(α)− l(γ) (here, l(β) denotes the level of β (cf. Remark

3.1.8)), then there exist α1 · · · , αm in JL such that Iα \ Iγ = {α1 · · · , αm}. Let


now β, β′ be two non-comparable elements in L. Let α = β ∨ β′, γ = β ∧ β′. Let

l(β)− l(γ) = r, l(β′)− l(γ) = s. Then there exist β1, · · · , βr, and β′1, · · · , β′s in JL

such that

Iβ \ Iγ = {β1, · · · , βr}

Iβ′ \ Iγ = {β′1, · · · , β′s}

Iα \ Iβ = {β′1, · · · , β′s}

Iα \ Iβ′ = {β1, · · · , βr}Hence we obtain

mβ = mγuβ1 · · · uβr

mβ′ = mγuβ′1 · · ·uβ′s

mα = mβuβ′1 · · ·uβ′s

mα = mβ′uβ1 · · ·uβr

From this it follows that

mαmγ = mβmβ′

Thus for each diamond in L, i.e., a quadruple (β, β′, β ∨ β′, β ∧ β′), we have

XβXβ′ − Xβ∨β′Xβ∧β′ is in the kernel of the surjective map F . Hence F factors

through K[XL]; hence, we obtain closed immersions (of irreducible varieties):

Y ↪→ XL ↪→ A#L

(Y being V(IA)). But dimension considerations imply that Y = XL (note that in

view of Proposition 4.1.4, dimY = d = dimXL (cf. Theorem 3.2.11)), and the

result follows.

As an immediate consequence, we obtain (as seen in the proof of the above

Lemma)

Theorem 4.1.7. We have an isomorphism K[XL] ∼= K[mA, A ∈ I(JL)].


Denote M := X(T ), the character group pf T . Let N = Z-dual of M , and

{ey, y ∈ JL} be the basis of N dual to {fz, z ∈ JL}. As above, for A ∈ I(JL), let

fA :=∑z∈A

fz

Let V = NR(= N⊗ZR). Let σ ⊂ V be the cone such that XL = Xσ. Let σ∨ ⊂ V ∗

be the cone dual to σ. Let Sσ = σ∨ ∩M , so that K[XL] equals the semi group

algebra K[Sσ].

As an immediate consequence of Theorem 4.1.7, we have

Proposition 4.1.8. The semigroup Sσ is generated by {fA, A ∈ I(JL)}.

Let M(JL) be the set of maximal elements in the poset JL. Let Z(JL) denote

the set of all covers in the poset JL (i.e., (z, z′), z > z′ in the poset JL, and there

is no other element y ∈ JL such that z > y > z′). For a cover (y, y′) ∈ Z(JL),

denote

vy,y′ := ey′ − ey

Proposition 4.1.9. The cone σ is generated by {ez, z ∈ M(JL), vy,y′ , (y, y′) ∈Z(JL)}.

Proof 4.1.10. Let θ be the cone generated by {ez, z ∈ M(JL), vy,y′ , (y, y′) ∈Z(JL)}. Then clearly any u in Sσ is non-negative on the generators of θ, and

hence σ∨ ⊆ θ∨. We shall now show that θ∨ ⊆ σ∨, equivalently, we shall show

that Sθ ⊆ Sσ. Let f ∈ M , say, f =∑

z∈JL

azfz. Then it is clear that f is in Sθ if

and only if

(∗) az ≥ 0, for z maximal in JL, and ax ≥ ay, for x, y ∈ JL, x < y

Claim: Let f ∈ M . Then f has property (*) if and only if f =∑

A∈I(JL)

cAfA,


cA ∈ Z+.

Note that Claim implies that Sθ ⊆ Sσ, and the required result follows.

Proof of Claim: The implication ⇐ is clear.

The implication ⇒: Let f ∈ Sθ, say, f =∑

z∈JL

azfz. The hypothesis that f

has property (*) implies that az being non-negative for z maximal in JL, ax is

non-negative for all x ∈ JL. Thus f =∑

z∈JL

azfz, az ∈ Z+, z ∈ JL. Further, the

property in (*) that ax ≥ ay if x < y implies that

{x | ax 6= 0}

is an ideal in JL. Call it A; in the sequel we shall denote A also by If . Let

m = min{ax, x ∈ A}. Then either f = mfA in which case the Claim follows, or

f = mfA +f1, where f1 is in Sθ (note that f1 also has property (*)); further, If1 is

a proper subset of A. Thus proceeding we arrive at positive integers m1, · · · ,mr,

elements f1, · · · , fr (in Sθ) with Ai := Ifi, an ideal in JL, and a proper subset of

Ai−1 := Ifi−1such that

f = mfA + m1fA1 + · · ·+ mrfAr

(In fact, at the last step, we have, fr = mrfAr .) The Claim and hence the required

result now follows.

4.1.11 Analysis of faces of σ

We shall concern ourselves just with the closed points in XL. So in the sequel,

by a point in XL, we shall mean a closed point. Let τ be a face of σ. Let Pτ be

the distinguished point of Oτ with the associated maximal ideal being the kernel


of the map

K[Sσ] → K,

u ∈ Sσ, u 7→

1, if u ∈ τ⊥

0, otherwise

Then for a point P ∈ XL (identified with a point in Al), denoting by P (α), the

α-th co-ordinate of P , we have,

Pτ (α) =

1, if fIα ∈ τ⊥

0, otherwise

With notation as above, let

Dτ = {α ∈ L |Pτ (α) 6= 0}

We have,

Lemma 4.1.12. Dτ is an embedded sublattice of L.

Proof 4.1.13. Let θ, δ be a pair of non-comparable elements in Dτ . The fact that

Pτ (θ), Pτ (δ) are non-zero, together with the diamond relation xθxδ = xθ∨δxθ∧δ

implies that

Pτ (θ∨ δ), Pτ (θ∧ δ) are again non-zero, and are equal to 1 (note that any non-zero

co-ordinate in Pτ is in fact equal to 1). Thus, Dτ is a sublattice of L.

The above reasoning implies that if θ, δ in Dτ form the main diagonal in a diamond

D in L, then Pτ (α), Pτ (β) are non-zero, and are equal to 1, where α, β form the

skew diagonal of D. Hence α, β are in D, and hence D ⊂ Dτ . Thus we obtain

that Dτ is an embedded sublattice of L.

Conversely, we have


Lemma 4.1.14. Let D be an embedded sublattice in L. Then D determines a

unique face τ of σ such that Dτ equals D.

Proof 4.1.15. Denote P to be the point in A#L with

P (α) =

1, if α ∈ D

0, otherwise

Then P is in L (since D is an embedded sublattice in L, the co-ordinates of P

satisfy all the diamond relations). Set

u =∑

α∈D

fIα , τ = σ ∩ u⊥

Then clearly, P = Pτ and Dτ = D

Thus in view of the two Lemmas above, we have a bijection

{ faces of σ} bij↔{ embedded sublattices of L}

Proposition 4.1.16. Let τ be a face of σ. Then we have Oτ = XDτ .

Proof 4.1.17. Recall (cf. Lemma 2.4.18) that K[Oτ ] = K[Sσ ∩ τ⊥]. From the

description of Pτ , we have that τ⊥ is the span of {fIα , α ∈ Dτ}; hence

(∗) Sσ ∩ τ⊥ = {∑

α∈Dτ

cαfIα , cα ∈ Z+}

Thus we obtain

(∗∗) Sσ ∩ τ⊥ = Sστ

(here, στ is as in Lemma 2.4.18). On the other hand, if η is the cone associated to

the toric variety XDτ , then by Proposition 4.1.8 we have that Sη is the semigroup

generated by {fIα , α ∈ Dτ}. Hence we obtain that η = στ (in view of (*) and

(**)). The required result now follows.

4.2. Generation of the cotangent space at Pτ 45

4.2 Generation of the cotangent space at Pτ

For α ∈ L, let us denote the image of Xα in RL (under the surjective map

K[Xα, α ∈ L] → RL) by xα. Let Rτ = K[X(Dτ )], the co-ordinate ring of

X(Dτ ), and πτ : RL → Rτ be the canonical surjective map induced by the closed

immersion X(Dτ ) ↪→ XL; clearly, kernel of πτ is generated by {xθ, θ ∈ L \ Dτ}.Set

Fα =

xα, if α 6∈ Dτ


Denoting by Mτ , the maximal ideal in RL corresponding to Pτ , we have (cf.

§4.1.11)

Lemma 4.2.1. The ideal Mτ is generated by {Fα, α ∈ L}.

A set of generators for the cotangent space Mτ/M2τ

For F ∈ Mτ , let F denote the class of F in Mτ/M2τ .

Lemma 4.2.2. Fix a maximal chain Γ in Dτ . For any β ∈ Dτ \ Γ, we have that

in Mτ/M2τ , F β is in the span of {F γ, γ ∈ Γ}

Proof 4.2.3. Fix a β ∈ Dτ \ Γ; denote by lτ (β), the level (cf. Remark 3.1.8) of β

considered as an element of the distributive lattice Dτ . We shall prove the result

by induction on lτ (β). If lτ (β) = 0, then β coincides with the (unique) minimal

element of Dτ , and there is nothing to prove. Let then lτ (β) ≥ 1. Let β′ in Dτ

be such that β covers β′. Then Iβ \ Iβ′ = {θ}, for an unique θ ∈ JDτ (cf. Lemma

3.1.24). Then there exists a unique cover (γ, γ′) in Γ such that Iγ \ Iγ′ = {θ} (cf.

Lemma 3.1.24).

Claim: Fβ − Fβ′ ≡ Fγ − Fγ′(modMτ ).


First observe that the fact that θ belongs to Iβ, Iγ and does not belong to Iβ′ , Iγ′

implies

(∗) γ′ 6≥ β, β′ 6≥ γ

We now divide the proof of the Claim into the following two cases.

Case 1: γ′ < β.

This implies that γ′ is in fact less than β′, and γ < β (since, Iβ \ Iβ′ = Iγ \ Iγ′ =

{θ}, and θ 6∈ Iγ′); this in turn implies that β′ 6≤ γ (for, otherwise, β′ ≤ γ would

imply γ′ < β′ < γ, not possible, since, (γ, γ′) is a cover). Let D be the diamond

having (γ, β′) as the skew diagonal. Now γ′ being less than both γ and β′, we get

that γ′ ≤ β′ ∧ γ, and in fact equals β′ ∧ γ (note that the fact that γ′ ≤ β′ ∧ γ < γ

together with the fact that (γ, γ′) is a cover in Dτ implies that γ′ = β′ ∧ γ). In a

similar way, we obtain that β = β′ ∨ γ. Now the diamond relation xβ′xγ = xβxγ′

implies the claim in this case (by definition of Fξ’s).

Case 2: γ′ 6≤ β.

This implies that γ 6≤ β.

If γ > β, then clearly

γ = γ′ ∨ β, β′ = γ′ ∧ β

Claim in this case follows as in Case 1.

Let then γ 6≥ β. Then we have that (γ, β), (γ′, β′) are non-comparable pairs (note

that γ′ ≤ β′ if and only if γ′ < β). Denote δ = γ ∧ β, δ′ = γ′ ∧ β′. The facts that

Iδ = Iγ ∩ Iβ, Iδ′ = Iγ′ ∩ Iβ′ , Iγ = Iγ′ ∪ {θ}, Iβ = Iβ′ ∪ {θ}

imply that

Iδ = Iδ′ ∪ {θ}


Hence we obtain

Iγ′ ∩ Iδ = Iγ′ ∩ Iδ′ = Iδ′ ; Iγ′ ∪ Iδ = Iγ

Iβ′ ∩ Iδ = Iβ′ ∩ Iδ′ = Iδ′ ; Iβ′ ∪ Iδ = Iβ

Thus we obtain that

γ′ ∧ δ = δ′, γ′ ∨ δ = γ

β′ ∧ δ = δ′, β′ ∨ δ = β

Considering the diamonds with skew diagonals (γ′, δ), (β′, δ)respectively, the

diamond relations

xγ′xδ = xγxδ′ , xβ′xδ = xβxδ′

imply that in Mτ/M2τ , we have the following relations

F γ − F γ′ = F δ − F δ′ ; F β − F β′ = F δ − F δ′

Hence we obtain that Fβ − Fβ′ ≡ Fγ − Fγ′(modMτ ) as required.

This completes the proof of the Claim. Note that Claim implies the required

result (by induction on lτ (β)). (Note that when lτ (β) = 1, then β′ is the (unique)

minimal element of Dτ , and hence β′ ∈ Γ. The result in this case follows from

the Claim).

Under the (surjective) map πτ : RL → Rτ , denote πτ (Mτ ) by M ′τ ; then πτ induces

a surjection πτ : Mτ/M2τ → M ′

τ/M′2τ .

Corollary 4.2.4. {πτ (F γ), γ ∈ Γ} is a basis for M ′τ/M

′2τ .

Proof 4.2.5. We have that dimM ′τ/M

′2τ ≥ dimX(Dτ ) = #Γ (cf. Corollary

3.2.12(1)). On the other hand, the above Lemma implies that dimM ′τ/M

′2τ ≤ #Γ,

and the result follows.

Lemma 4.2.6. Let α ∈ L \ Dτ be such that there exists an element β ∈ Dτ and

a diamond D in L such that


(1) (α, β) is a diagonal (main or skew) in D.

(2) D ∩Dτ = {β}.

Then Fα ∈ M2τ .

Proof 4.2.7. Let us denote the remaining vertices of the diamond by θ, δ. Writing

the diamond relation xαxβ = xθxδ in terms of the Fξ’s, we have,

xαxβ − xθxδ = Fα(1− Fβ)− FθFδ

Hence in RL, we have,

Fα = FαFβ + FθFδ

The required result follows from this.

The set Eτ

Define

Eτ := {α ∈ L, α as in Lemma 4.2.6}

In the sequel, we shall refer to an element α ∈ Eτ as an Eτ -element.

The equivalence relation:

For two distinct elements θ, δ ∈ L \ Dτ , we say θ is equivalent to δ (and denote

it as θ ∼ δ) if there exists a sequence θ = θ1, · · · , θn = δ in L \ Dτ such that

(θi, θi+1) forms one side of a diamond in L whose other side is contained in Dτ .

For θ ∈ L \ Dτ , we shall denote by [θ], the set of all elements of L \ Dτ equivalent

to θ, if there exists such a diamond as above having θ as one vertex; if no such

diamond exists, then [θ] will denote the singleton set {θ}. Clearly, for all θ in a

given equivalence class, Fθ (mod M2τ ) is the same (by consideration of diamond

relations), and we shall denote it by F [θ] or also just F θ. Note also that in view


of Lemma 4.2.6, F θ = 0 in Mτ/M2τ , if [θ] ∩ Eτ 6= ∅. We shall refer to [θ] as a

Eτ -class or a non Eτ -class according as [θ] ∩ Eτ is non-empty or empty.

The sublattice Λτ (Γ): Fix a chain Γ in Dτ . Let Λτ (Γ) denote the union of all

the maximal chains in L containing Γ. Note that α in L is in Λτ (Γ) if and only

if α is comparable to every γ ∈ Γ.

Lemma 4.2.8. Λτ (Γ) is a sublattice of L.

Proof 4.2.9. Let (θ, δ) be a pair of non-comparable elements in Λτ (Γ). Then for

any γ ∈ Γ, we have (by definition of Λτ (Γ)) that either γ is less than both θ and

δ or greater than both θ and δ; in the former case, γ is less than both θ ∨ δ and

θ ∧ δ, and in the latter case, γ is greater than both θ ∨ δ and θ ∧ δ.

Remark 4.2.10. Λτ (Γ) need not be an embedded sublattice:

Take L to be the distributive lattice consisting of {(i, j), 1 ≤ i, j ≤ 4} with the

partial order (a1, a2) ≥ (b1, b2) ⇔ ar ≥ br, r = 1, 2. Take

Dτ := {(i, j), 2 ≤ i, j ≤ 3}, Γ := {(2, 2), (3, 2), (3, 3)}

Then

Λτ (Γ) = Γ ∪ I1 ∪ I2

where I1 = {(1, 1), (1, 2), (2, 1)}, I2 = {(3, 4), (4, 3), (4, 4)}. Consider x =

(2, 4), y = (4, 2); then x ∨ y = (4, 4), x ∧ y = (2, 2). Now x ∨ y, x ∧ y are in

Λτ (Γ), but x, y are not in Λτ (Γ). Thus Λτ (Γ) is not an embedded sublattice.

Proposition 4.2.11. Let θ0 ∈ L \ Dτ . Then θ0 ∼ µ, for some µ ∈ Λτ (Γ) ∪ Eτ .

Proof 4.2.12. Let us denote L′τ = L \ Dτ . By induction, let us suppose that

the result holds for all θ ∈ L′τ , θ > θ0; we shall see that the proof for the case


when θ0 is a maximal element in L′τ (the starting point of induction) is included

in the proof for a general θ0.

If θ0 ∈ Λτ (Γ), then there is nothing to prove. Let then θ0 6∈ Λτ (Γ). Fix γ1 minimal

in Γ such that θ0 and γ1 are non-comparable. Let ξ = γ1 ∧ θ0, θ1 = γ1 ∨ θ0. We

divide the proof into the following two cases.

Case 1: γ1 is the minimal element of Γ (note that γ1 is the minimal element of

Dτ also).

We have ξ ∈ Λτ (Γ) (since ξ < γ1); further, ξ 6∈ Dτ (again, since ξ < γ1, the

minimal element of Dτ ).

Subcase 1(a): Let θ1 be in Dτ . Then considering the diamond with

(θ1, γ1), (θ0, ξ) as opposite sides, we have, θ0 ∼ ξ, and the result follows (note

that ξ ∈ Λτ (Γ)).

Subcase 1(b): Let θ1 6∈ Dτ . This implies θ0 ∈ Eτ , and the result follows.

Case 2: γ1 is not the minimal element of Γ.

Let γ0 ∈ Γ be such that γ0 is covered by γ1 in Dτ . Then ξ ≥ γ0 (since, both θ0

and γ1 are > γ0).

Subcase 2(a): Let ξ > γ0. Then the fact that (γ1, γ0) is a cover in Dτ together

with the relation γ0 < ξ < γ1 implies that ξ ∈ Λτ (Γ) \ Γ; in particular, ξ 6∈ Dτ .

Then as in Case 1, we obtain that θ0 ∈ Eτ if θ1 6∈ Dτ , and θ0 ∼ ξ, if θ1 ∈ Dτ ; and

the result follows (again note that ξ ∈ Λτ (Γ)).

Subcase 2(b): Let ξ = γ0. We first note that θ1 6∈ Dτ ; for θ1 ∈ Dτ would imply

that θ0 ∈ Dτ (since Dτ is an embedded sublattice (cf. Lemma 4.1.12)). Hence by

induction hypothesis, we obtain that

(∗) θ1 ∼ η, for some η ∈ Λτ (Γ) ∪ Eτ

Also, considering the diamond with (θ1, θ0), (γ1, γ0) as opposite sides, we have,


θ0 ∼ θ1. Hence, the result follows in view of (*).

Note that the above proof includes the proof of the starting point of induction,

namely, θ0 is a maximal element in L′τ . To make this more precise, let γ1, ξ, θ1

be as above. Proceeding as above, we obtain (by maximality of θ0) that θ1 ∈ Dτ .

Hence Subcases 1(b) and 2(b) do not exist (since in these cases θ1 6∈ Dτ ). In

Subcases 1(a) and 2(a), we have ξ ∈ Λτ (Γ), and θ0 ∼ ξ. The result now follows

in this case.

Lemma 4.2.13. Let (θ1, δ1), (θ2, δ2) be two skew pairs in Λτ (Γ) such that either

θ1 ∨ δ1 = θ2 ∨ δ2 or θ1 ∧ δ1 = θ2 ∧ δ2, say θ1 ∨ δ1 = θ2 ∨ δ2, and γi := θi ∧ δi, i = 1, 2

are in Γ. Then γ1 = γ2.

Proof 4.2.14. Assume γ1 6= γ2. We have γ1, γ2 are comparable (since Γ is totally

ordered), say γ1 > γ2. Denote α := θ1∨δ1 = θ2∨δ2. The fact that any element of

Λτ (Γ) is comparable to all elements of Γ together with the hypothesis that (θ2, δ2)

is a skew pair implies that either θ2, δ2 are both > γ1 or are both < γ1. If θ2, δ2

are both > γ1, then we would obtain γ2(= θ2 ∧ δ2) ≥ γ1 which is not true. Hence

we obtain that θ2, δ2 are both < γ1; this implies that α(= θ2 ∨ δ2) < γ1 which is

again not true. Hence our assumption is wrong and the result follows.

Let Yτ (Γ) = Λτ (Γ) \ Eτ . Let us write

Yτ (Γ) = Γ∪Zτ (Γ)

Gτ (Γ) = {[θ], θ ∈ Zτ (Γ), [θ] is a non Eτ−class}

Combining the above Proposition with Lemmas 4.2.2,4.2.6, we obtain

Proposition 4.2.15. Mτ/M2τ is generated by {F [θ], [θ] ∈ Gτ (Γ)}∪{F γ, γ ∈ Γ}.

4.3. A basis for the cotangent space at Pτ 52

4.3 A basis for the cotangent space at Pτ

In this section, we shall show that {F [θ], [θ] ∈ Gτ (Γ)}∪{F γ, γ ∈ Γ} is in fact a

basis for Mτ/M2τ . We first recall some basic facts on tangent cones and tangent

spaces.

Let X = Spec R ↪→ Al be an affine variety; let S be the polynomial algebra

K[X1, · · · , Xl]. Let P ∈ X, and let MP be the maximal ideal in K[X]

corresponding to P (we are concerned only with closed points of X). Let A =

OX,P , the stalk at P ; denote the unique maximal ideal in A by M(= MP RMP).

Then Spec gr(A,M), where gr(A,M) = ⊕j∈Z+

M j/M j+1 is the tangent cone to X

at P , and is denoted TCP X. Note that

gr(R, MP ) = gr(A,M)

Tangent cone & tangent space at P : Let I(X) be the vanishing ideal of X

for the embedding X = SpecR ↪→ Al. Expanding F ∈ I(X) in terms of the local

co-ordinates at P , we have the following:

• TP (X), the tangent space to X at P is the zero locus of the linear forms of F ,

for all F ∈ I(X), i.e, the degree one part in the (polynomial) local expression for

F .

• TCP (X), the tangent cone to X at P is the zero locus of the initial forms (i.e.,

form of smallest degree) of F , for all F ∈ I(X).

In the sequel, we shall denote the initial form of F by IN(F ).

We collect below some well-known facts on singularities of X:

Facts: 1. dimTP X ≥ dim X with equality if and only if X is smooth at P .

2. X is smooth at P if and only if gr(R,MP ) is a polynomial algebra.


4.3.1 Determination of the degree one part of J(τ):

We now take X = XL, P = Pτ ; we shall denote I = I(XL),MP = Mτ . As above,

for F ∈ Mτ , let F denote the class of F in Mτ/M2τ . Let J(τ) be the kernel of the

surjective map

fτ : K[Xθ, θ ∈ L] → gr(R,Mτ ), Xθ 7→ Fθ

For r ∈ N, let f (r) be the restriction of f to the degree r part of the polynomial

algebra K[Xθ, θ ∈ L]. We are interested in

f (1)τ : ⊕

θ∈LKXθ → Mτ/M

2τ

We shall first describe a complement to the kernel of f(1)τ , and then deduce a basis

for Mτ/M2τ (which will turn out to be the set {F [θ], [θ] ∈ Gτ (Γ)}∪{F γ, γ ∈ Γ}).

Let Vτ be the span of {Fβ, β ∈ Dτ} (in Mτ/M2τ ); by Lemmas 4.2.2, we have that

Vτ is spanned by {F γ, γ ∈ Γ}. Let Wτ be the span of {Fθ, θ 6∈ Dτ} (in Mτ/M2τ ).

Lemma 4.3.2. {F γ, γ ∈ Γ} is a basis for Vτ .

Proof 4.3.3. The surjective map πτ : Mτ/M2τ → M ′

τ/M′2τ induces a surjection

Vτ → M ′τ/M

′2τ , and the result follows from Corollary 4.2.4.

Lemma 4.3.4. The sum Vτ + Wτ is direct.

Proof 4.3.5. Let v ∈ Vτ ; by Lemma 4.3.2, we can write v =∑γ∈Γ

aγFγ. Now if

v ∈ Wτ , then under the map

πτ : Mτ/M2τ → M ′

τ/M′2τ

we obtain that πτ (v) = 0 (since, Wτ ⊆ ker πτ ). Hence we obtain that

πτ (∑γ∈Γ

aγFγ) = 0 in M ′τ/M

′2τ ; this in turn implies that aγ = 0,∀γ (cf. Corollary

4.2.4). Hence v = 0 and the required result follows.


Let us write ⊕θ∈L

KXθ = Aτ ⊕Bτ where

Aτ = ⊕β∈Dτ

KXβ, Bτ = ⊕θ 6∈Dτ

KXθ

Write f(1)τ = g

(1)τ + h

(1)τ , where g

(1)τ (respectively h

(1)τ ) is the restriction of f

(1)τ to

Aτ (respectively Bτ ). Note that we have surjections

g(1)τ : Aτ → Vτ , h(1)

τ : Bτ → Wτ

As a consequence of Lemma 4.3.4, we get the following

Corollary 4.3.6. ker f(1)τ = ker g

(1)τ ⊕ ker h

(1)τ

Proof 4.3.7. The inclusion“⊇” is clear. Let then v ∈ ker f(1)τ ; write v = a + b,

where a ∈ Aτ , b ∈ Bτ . Then denoting a′ = f(1)τ (a), b′ = f

(1)τ (b), we have, 0 = a′+b′.

Also, we have that a′ ∈ Vτ , b′ ∈ Wτ . Hence in view of Lemma 4.3.4, we obtain

a′ = 0 = b′. This implies that a ∈ ker g(1)τ , b ∈ ker h

(1)τ , and the result follows.

Lemma 4.3.8. The span of {Xγ, γ ∈ Γ} is a complement to the kernel of g(1)τ .

Proof 4.3.9. We have, g(1)τ (Xβ) = Fβ. By Lemma 4.3.2, we have that {Fγ, γ ∈ Γ}

is a basis for Vτ . The result now follows.

Let {ξ1, · · · , ξr} be a complete set of representatives for Gτ (Γ).

Lemma 4.3.10. The span of {Xξ1 , · · · , Xξr} is a complement to the kernel of

h(1)τ .

Proof 4.3.11. A typical element F of I(XL) such that IN( F ) is in Bτ is of the

form

F = F1 + F2

where F1 is a linear sum of diamond relations arising from diamonds having

precisely one vertex in Dτ , and F2 is a linear sum of diamond relations arising


from diamonds having precisely one side in Dτ . Note that in a typical term in F1,

the linear term is of the form aαXα, aα ∈ K where α ∈ Eτ ; similarly, in a typical

term in F2, the linear term is of the form bθδ(Xθ −Xδ), θ, δ 6∈ Dτ , θ ∼ δ, bθδ ∈ K.

Hence the kernel of h(1)τ is generated by {Xα, α ∈ Eτ} ∪ {(Xθ −Xδ), θ ∼ δ}. The

required result now follows in view of Proposition 4.2.11.

Theorem 4.3.12. {F [θ], [θ] ∈ Gτ (Γ)}∪{F γ, γ ∈ Γ} is a basis for Mτ/M2τ .

Proof 4.3.13. In view of Lemmas 4.3.8, 4.3.10 and Corollary 4.3.6, we obtain

that

⊕γ∈Γ

KXγ ⊕ ⊕1≤i≤r

KXξiis a complement to the kernel of the surjective map

f(1)τ : ⊕

θ∈LKXθ → Mτ/M

2τ . The result now follows.

Let TτXL denote the tangent space to XL at Pτ . As an immediate consequence

of the above Theorem, we have the following

Corollary 4.3.14. dimTτXL = #Gτ (Γ) + #Γ.

In view of the above Corollary, we obtain that XL is singular along Oτ if and only

if #Gτ (Γ) + #Γ > #L. Let

SL = {τ < σ |#Gτ (Γ) + #Γ > #L}

(here, σ is the cone associated to XL). We obtain from Proposition 4.1.16 the

following

Theorem 4.3.15. Sing XL = ∪τ∈SL

X(Dτ ).

Chapter 5

Standard Basis for Tangent Cone

5.1 Analysis of diamond relations:

For a diamond D in L, with (θ, δ) as the skew diagonal, let

fD = XθXδ −Xθ∨δXθ∧δ

We shall refer to fD as a diamond relation.

The ideal b(τ): Let b(τ) be the ideal in K[Xα, α ∈ L] generated by ker π(1) and

the diamond relations fD′ for D′ in Λτ (Γ).

Our main goal in this section is to show that given a diamond D in L, IN fD

is in b(τ). Our first step is to collect IN fD, for all diamonds D in L. In our

discussion in this section, we will be repeatedly using the following two lemmas.

Lemma 5.1.1. Let D be a diamond with diagonals (θ1, θ4), (θ2, θ3). Suppose

either

Fθ1 ≡ Fθ2(modM2τ ), Fθ3 ≡ Fθ4(modM2

τ )

or


τ )

56

5.1. Analysis of diamond relations: 57

Then fD ∈ b(τ)

Proof 5.1.2. Let us prove the case when Fθ1 ≡ Fθ3(modM2τ ), Fθ2 ≡

Fθ4(modM2τ )) (proof of the other case being similar). Let (θ2, θ3) be the skew

diagonal. Hypothesis implies that Xθ1 − Xθ3 , Xθ2 − Xθ4 are in ker π(1). The

assertion follows by writing

fD(= ±(Xθ2Xθ3 −Xθ1Xθ4)) = ±[Xθ3(Xθ2 −Xθ4)−Xθ4(Xθ1 −Xθ3))]

Lemma 5.1.3. Let D1, D2 be two diamonds with diagonals

{(θ1, θ4), (θ2, θ3)}, {(δ1, δ4), (δ2, δ3)} respectively; further, let (θ2, θ3), (δ2, δ3) be

the skew diagonals for D1, D2 respectively. Suppose Fθj≡ Fδj

(modM2τ ).

Then fD1 ≡ fD2(mod b(τ)).

Proof 5.1.4. The assertion follows by writing fD1(= Xθ2Xθ3 −Xθ1Xθ4) as

fD1 = Xθ3(Xθ2 −Xδ2)−Xθ4(Xθ1 −Xδ1)−Xδ1(Xθ4 −Xδ4) + Xδ2(Xθ3 −Xδ3) + fD2

(note that hypothesis implies that Xθj−Xδj

, j = 1, 2, 3, 4 are in ker π(1)).

As mentioned in the beginning of this section, we want to collect IN fD, for all

diamonds D in L. Since we know the description of Mτ/M2τ (and hence ker π(1)),

we shall suppose the following:

(i) In view of Lemma 4.1.12 and Lemma 4.2.4, we may suppose that any diagonal

of D is not contained in Dτ .

(ii) In view of Lemma 4.2.2, we may suppose that any side of D is not contained

in Dτ (note that in this case, in view of that Lemma, IN fD ∈ ker π(1)).

In view of (i), (ii), we may suppose that #{D ∩Dτ} ≤ 1.

(iii) Let #{D∩Dτ} = 1; let β be the vertex of D such that β ∈ Dτ . Let α be the

other vertex on the diagonal of D through β. Let (θ, δ) be the other diagonal of D.


We obtain that α ∈ Eτ (note that θ, δ 6∈ Dτ ), and hence IN fD = Xα(∈ ker π(1))

(cf. Lemma 4.2.6).

Thus we may suppose that D ∩Dτ 6= ∅.

(iv) In view of Lemma 4.2.6, we may suppose that any side of D is not contained

in Eτ (note that in this case, in view of that Lemma, fD ∈ b(τ)). Thus we may

suppose that # D ∩ Eτ ≤ 2.

For the rest of this section, D will denote a diamond in L satisfying

D ∩Dτ = ∅, # D ∩ Eτ ≤ 2

In particular, fD is homogeneous (and hence IN fD = fD).

Let b(τ) be as above (namely, the ideal in K[Xα, α ∈ L] generated by ker π(1)

and the diamond relations fD′ for D′ in Λτ (Γ)). Let us denote Γ by

Γ = {γn+1 > · · · > γ1}

where n is the rank of the distributive lattice Dτ . Denote γ0 := 0 (the “zero

element” of L). For 0 ≤ i ≤ n + 1, let

Λi(Γ) := {θ ∈ L | (θ, γr) comparable, r ≤ i}

We have

Λi(Γ) ⊃ Λi+1(Γ), Λn+1(Γ) = Λτ (Γ), Λ0(Γ) = L

(Note that if θ ∈ L \ Λτ (Γ), then θ ∈ Λi(Γ) where i + 1 is the smallest such that

(θ, γi+1) is skew.)

Lemma 5.1.5. Λi(Γ) is a sublattice of L.

Proof 5.1.6. Let (τ, ϕ) be a skew pair in Λi(Γ). Then for 1 ≤ r ≤ i, we have

that γr is comparable to both τ, ϕ. In fact, γr is either less than both τ, ϕ or


greater than both τ, ϕ; hence, τ ∨ ϕ, τ ∧ ϕ are both comparable to γr, and we

obtain that τ ∨ ϕ, τ ∧ ϕ are in Λi(Γ).

Let D be a diamond in Λi(Γ) for some i, 0 ≤ i ≤ n. Let bi+1(τ) be the ideal

in K[Xα, α ∈ L] generated by ker π(1) and the diamond relations fD′ for D′ in

Λi+1(Γ).

Proposition 5.1.7. fD is in bi+1(τ).

Before proving the Proposition, let us derive the following important consequence

(which is in fact the essence of this section as mentioned above).

Corollary 5.1.8. fD is in b(τ).

Proof 5.1.9. This is immediate from Proposition 5.1.7 by decreasing induction

on i; the starting point being i = n + 1 for which the result is clear

(note that fD ∈ bn+1(τ)(= b(τ))).

Proof of Proposition 5.1.7: Let us denote the main diagonal of D by (θ4 > θ1),

and the skew diagonal by (θ2, θ3). We break our discussion into the two cases: D

contained in Λi(Γ) \ Λi+1(Γ) and D not contained in Λi(Γ) \ Λi+1(Γ).

A. Let D be contained in Λi(Γ) \ Λi+1(Γ). Denote γ := γi+1. We have

(θj, γ), j = 1, 2, 3, 4 are skew. Denote

εj := θj ∨ γ, δj := θj ∧ γ, j = 1, 2, 3, 4

By distributivity of join, meet operations, we have

ε2 ∨ ε3 = ε4, ε2 ∧ ε3 = ε1

δ2 ∨ δ3 = δ4, δ2 ∧ δ3 = δ1


For simplicity, let us refer to the diamond with εj (respectively δj) as vertices

as the ε-diamond (respectively δ-diamond), whenever (ε2, ε3) (resp. (δ2, δ3)) is

skew. Note that εj, δj, j = 1, 2, 3, 4 belong to Λi+1(Γ); for, (θj, γ) being skew,

j = 1, 2, 3, 4, for any γr, r ≤ i, we have that either both θj, γ are > γr or both

< γr. Thus for r ≤ i, εj, γr (respectively δj, γr) are comparable, and in addition

εj, γ (respectively δj, γ) are comparable (note that γ = γi+1).

Also, the fact that θj 6∈ Dτ implies (by consideration of the diamond having (θj, γ)

as the skew diagonal):

(∗) At most one of {εj, δj} is in Dτ , j = 1, 2, 3, 4

(since Dτ is an embedded sublattice of L).

Case 1: Let D ∩Eτ = ∅. This implies (by (*)) that precisely one of {εj, δj} is in

Dτ for j = 1, 2, 3, 4 (by definition of Eτ ). If either one of the pairs (ε1, ε4), (ε2, ε3)

is contained in Dτ , then εj ∈ Dτ for j = 1, 2, 3, 4.

(This is clear if ε2, ε3 are comparable - for then {ε2, ε3} = {ε1, ε4}; if (ε2, ε3) is

skew, then this follows from the fact that Dτ is an embedded sublattice of L).

As a consequence we obtain (by considering the diamond with (θj, γ) as the skew

diagonal)

Fθj≡ Fδj

(modM2τ ), j = 1, 2, 3, 4

If δ2, δ3 are comparable, say δ2 > δ3, then δ2 = δ4,δ3 = δ1. Hence we obtain

Fθ2 ≡ Fθ4(modM2τ ) Fθ3 ≡ Fθ1(modM2

τ )

It follows (in view of Lemma 5.1.1) that fD ∈ b(τ)).

If (δ2, δ3) is skew, then we obtain (in view of Lemma 5.1.3) that fD ≡fD′(mod b(τ)) where D′ is the δ-diamond (⊆ Λi+1). Hence it follows that

fD ∈ bi+1(τ).


Interchanging the roles of the ε’s and δ’s, we obtain that if either one of the pairs

(δ1, δ4), (δ2, δ3) is contained in Dτ , then we obtain the following:

(i) fD ∈ b(τ), if ε2, ε3 are comparable.

(ii) If (ε2, ε3) is skew, then fD ≡ fD′(mod b(τ)) where D′ is the ε-diamond

(⊆ Λi+1), and fD ∈ bi+1(τ).

To discuss the remaining possibilities, we shall denote the diagonals of D by

(θj, θm), (θk, θl). In view of the fact that precisely one of {εh, δh} is in Dτ for

h = j, k, l, m, we are left to discuss the case when the following hold:

(a) precisely one of the pairs (εj, εk), (εl, εm) is contained in Dτ and the other pair

has empty intersection with Dτ .

(b) precisely one of the pairs (δj, δk), (δl, δm) is contained in Dτ and the other pair

has empty intersection with Dτ .

Let {εj, εk} be contained in Dτ . This implies that {δl, δm} is contained in Dτ . We

have

(‡) Fδj≡ Fδk

(modM2τ ), Fεl

≡ Fεm(modM2τ )

(This is clear if (εj, εm) (resp. (δj, δm)) is non-skew in which case εj = εk, εl = εm

(resp. δj = δk, δl = δm); if (εj, εm) (resp. (δj, δm)) is skew, then this follows

by consideration of the ε-diamond (resp. δ-diamond)). Also considering the

diamonds with (θr, γ), r = j, k, l, m as the skew diagonals, we have

(‡‡) Fθh≡ Fδh

(modM2τ ), h = j, k Fθh

≡ Fεh(modM2

τ ), h = l,m

Hence we obtain (from (‡), (‡‡))

Fθj≡ Fθk

(modM2τ ), Fθl

≡ Fθm(modM2τ )

From this it follows (in view of Lemma 5.1.1) that fD is in bi+1(τ).


This completes the discussion in the case D ∩ Eτ = ∅.

Case 2: Let #{D ∩ Eτ} = 1.

As in Case 1, let us denote the diagonals of D by (θj, θm), (θk, θl). Let θj ∈ Eτ

(and hence the other vertices are not in Eτ ). As in Case 1, we may suppose that

none of the pairs (εk, εl), (εj, εm), (δk, δl), (δj, δm) is contained in Dτ . Also, since

θh 6∈ Eτ , h 6= j, we obtain that precisely one of {εh, δh}, h 6= j is in Dτ . Hence,

precisely one of the pairs in {(εk, εm), (εl, εm), (δk, δm), (δl, δm)} is contained in Dτ ,

say, {εk, εm} is contained in Dτ ; this implies

Fεl≡ Fεj

(modM2τ )

(This is clear if εj, εm are comparable - for then εj = εl εk = εm}; if (εj, εm) is

skew, then this follows by the consideration of the ε-diamond and the fact that

εk, εm are in Dτ .) This together with the consideration of the diamonds with

(θl, γ), (θj, γ) as skew diagonals implies

(1) Fθl≡ Fεl

≡ Fεj≡ Fθj

(modM2τ )

Also, we have, δl ∈ Dτ (since precisely one of {εl, δl} is in Dτ and εl 6∈ Dτ ). Hence

we obtain the following:

If δj ∈ Dτ , then

(2) Fθk≡ Fδk

≡ Fδm ≡ Fθm(modM2τ )

(Note that if δj, δm are comparable, then δk = δm, δj = δl; if (δj, δm) is skew then

Fδk≡ Fδm ≡ Fθm(mod M2

τ )). It follows from (1), (2) and Lemma 5.1.1 that fD is

in bi+1(τ).

If δj 6∈ Dτ , then

δk ∈ Eτ , Fθk≡ Fδk

(modM2τ )


(note that δl ∈ Dτ , δm 6∈ Dτ ). We have Xθj, Xδk

are in ker π(1) (since θj, δk ∈ Eτ ),

and Xθk−Xδk

(and therefore Xθk) ∈ ker π(1). Hence it follows that fD ∈ bi+1(τ).

Case 3: Let #{D ∩ Eτ} = 2.

This implies (in view of (iv) above) that a diagonal of D is contained in Eτ .

As in Case 1, let us denote the diagonals of D by (θj, θm), (θk, θl), and let us

suppose that {θj, θm} is contained in Eτ . This together with the hypothesis that

#{D ∩ Eτ} = 2 implies that θk, θl 6∈ Eτ , and hence precisely one of {εk, δk}and one of {εl, δl} is in Dτ . Also, as in Case 1, we may suppose that none of

the pairs (εk, εl), (εj, εm), (δk, δl), (δj, δm) is not contained in Dτ . Hence we obtain

that precisely one of the pairs (εk, δl), (εl, δk) is contained in Dτ and the other has

empty intersection with Dτ ; say {εk, δl} is contained in Dτ . In view of the fact

that at least one of {εj, εm} is not in Dτ , we have the following possibilities:

If both εj, εm are not in Dτ , then (εj, εm) is skew necessarily (since εk ∈ Dτ ), and

we have

εl ∈ Eτ , Fθl≡ Fεl

(modM2τ )

If εm ∈ Dτ , εj 6∈ Dτ , then Fθm ≡ Fδm(modM2τ ). Further, we have

if δj 6∈ Dτ , then δk ∈ Eτ , Fθk≡ Fδk

(modM2τ )

if δj ∈ Dτ , then Fθk≡ Fδk

≡ Fδm ≡ Fθm(modM2τ )

(Note that if δj 6∈ Dτ , then (δj, δm) is skew necessarily (since δm 6∈ Dτ , δl ∈ Dτ ).

If δj ∈ Dτ , then Fδk≡ Fδm ≡ Fθm(modM2

τ ) whether (δj, δm) is skew or not). It

follows that fD ∈ bi+1(τ).

The discussion of the case εj ∈ Dτ , εm 6∈ Dτ is completely analogous (the roles of j

and m are interchanged) in the above discussion and we obtain that fD ∈ bi+1(τ).

This finishes the proof of the Proposition in case A.


B. Let D be not contained in Λi(Γ) \ Λi+1(Γ). As above, denote γ := γi+1.

Let us denote the skew diagonal of D by (θ2, θ3), and the main diagonal by

(θ4 > θ1). The hypothesis implies that at least one of {θ2, θ3} 6∈ Λi+1(Γ) (note

that if both {θ2, θ3} are in Λi+1(Γ), then Lemma 5.1.5 would imply that D is

contained in Λi+1(Γ), and the result follows). We divide the discussion into two

cases:

(1) precisely one of {θ2, θ3} is in Λi+1(Γ)

(2) both θ2, θ3 are not in Λi+1(Γ).

CaseB1: Precisely one of {θ2, θ3} is in Λi+1(Γ), say, θ3 ∈ Λi+1(Γ) (the proof

for the case θ2 ∈ Λi+1(Γ) being similar). This implies that θ3, γ(= γi+1) are

comparable, and the pair (θ2, γ) is skew. We shall prove the case when θ3 < γ

(the proof for the case θ3 > γ being similar). We have θ1 < γ; further

δ3 = θ3, δ1 = θ1, ε1 = ε3 = γ

(by the definition of the δi’s and the εi’s).

We will be using the following four facts:

Fact 1: ε4 = ε2

(This follows, since, ε4 = ε3 ∨ ε2 = γ ∨ ε2 = ε2.)

Fact 2: If δ2, δ3 are comparable, then δ2 < δ3(= θ3) necessarily (since δ2 < θ2,

and (θ2, θ3) is skew), and we have

δ4 = δ3(= θ3), δ2 = δ1(= θ1)

(This follows since δ4 = δ3 ∨ δ2 = δ3. Similarly we have δ2 = δ1).

Fact 3: If θ4, γ are comparable, then γ < θ4 necessarily, and we have

δ4 = γ, θ4 = ε4


(This follows since θ4 < γ would imply that θ2 < γ, which is not true.)

Fact 4: If δ4 6∈ Dτ , then (θ4, γ) is skew necessarily.

(This follows since θ4, γ are comparable would imply (cf. Fact 3) that δ4 ∈ Dτ .)

We now divide the discussion into the following two subcases:

Subcase B1(a): Let δ2 ∈ Dτ .

By considering the diamond with (θ2, γ) as the skew diagonal we obtain that

ε2 6∈ Dτ ; hence

(∗) Fθ2 ≡ Fε2(modM2τ )

Also Fact 2 implies that (δ2, δ3) is skew necessarily (since δ1(= θ1) 6∈ Dτ ).

If δ4 6∈ Dτ , then considering the δ-diamond, and the diamond with (θ4, γ) as skew

diagonal (note that (θ4, γ) is skew (cf. Fact 4)), we obtain

θ3(= δ3), θ4 ∈ Eτ

(note that δ2 ∈ Dτ and therefore ε4(= ε2) 6∈ Dτ )). It follows that fD is in

fD ∈ b(τ) (since Xθ3 , Xθ4 are in ker π(1)).

Let then δ4 ∈ Dτ .

Claim 1: Fθ2 ≡ Fθ4(mod M2τ ).

If θ4, γ are comparable, then θ4 = ε4 (cf. Fact 3). The Claim follows from this in

view of (*) (note that ε4 = ε2 by Fact 1).

If (θ4, γ) is skew, then

(∗∗) Fθ4 ≡ Fε4(modM2τ )

(note that δ4 ∈ Dτ ). Claim 1 follows from (*)and (**) (in view of Fact 1).

Now in view of Fact 2 and the hypothesis that δ4 ∈ Dτ , we have that (δ2, δ3) is


skew; hence by considering the δ-diamond, we have

(∗ ∗ ∗) Fθ1 ≡ Fθ3(mod M2τ )

(note that δ1 = θ1, δ3 = θ3, and δ2, δ4 ∈ Dτ ). It follows from Claim 1, (***), and

Lemma 5.1.1 that fD ∈ b(τ).

Subcase B1(b): Let δ2 6∈ Dτ .

(i) Let ε2(= ε4) ∈ Dτ . Then we get

(†) Fθ2 ≡ Fδ2(modM2τ )

Also, we have (cf. Fact 1)

(††) δ1 = θ1, δ3 = θ3

The hypothesis that ε4 ∈ Dτ implies that (θ4, γ) is skew (cf. Fact 3), and hence

we obtain

(† † †) Fθ4 ≡ Fδ4(modM2τ )

If δ2, δ3 are comparable, then θ3 = δ4, θ1 = δ2 (cf. Fact 2), and we obtain (by

(†), (††), († † †))Fθ2 ≡ Fθ1(modM2

τ ), Fθ4 ≡ Fθ3(modM2τ )

It follows that fD ∈ b(τ) (cf. Lemma 5.1.1).

If (δ2, δ3) is skew, then by considering the δ-diamond, we obtain in view of

(†), (††), († † †), and Lemma 5.1.3 that fD ∈ bi+1(τ).

(ii) Let ε2(= ε4) 6∈ Dτ . This implies (by considering the diamond with (θ2, γ) as

the skew diagonal)

(¦) θ2 ∈ Eτ


(note that by hypothesis we have δ2 6∈ Dτ ).

If θ1 ∈ Eτ , then clearly fD ∈ b(τ).

Let then θ1 6∈ Eτ .

Claim 2: θ4 ∈ Eτ .

If δ2, δ3 are comparable, then (θ4, γ) is skew necessarily (since in this case

δ4(= θ3) 6∈ Dτ (cf. Fact 2)), and θ4 ∈ Eτ (by considering the diamond with

(θ4, γ) as skew diagonal, and also noting that by hypothesis ε4 6∈ Dτ ).

If (δ2, δ3) is skew, then by considering the δ-diamond we obtain that δ4 6∈ Dτ

(since δ1(= θ1) 6∈ Eτ ). Hence we obtain that (θ4, γ) is skew necessarily, and

therefore θ4 ∈ Eτ (note that by hypothesis ε4 6∈ Dτ ).

Thus Claim 2 follows. Claim 2 together with (¦) implies that fD ∈ b(τ).

This completes the discussion in Case B1.

Case B2: θ2, θ3 6∈ Λi+1(Γ).

This implies that at least one of {θ4, θ1} is in Λi+1(Γ). We divide the discussion

into the following two cases:

(1) precisely one of {θ4, θ1} is in Λi+1(Γ)

(2) both θ4, θ1 are in Λi+1(Γ).

Subcase B2(a): Precisely one of {θ4, θ1} is in Λi+1(Γ), say, θ4 ∈ Λi+1(Γ) (the

proof for the case θ1 ∈ Λi+1(Γ) being similar). Thus θ4, γ are comparable, and

the pair (θ1, γ) is skew.

We first gather some facts:

Fact 5: We have θ4 > γ necessarily (since (θi, γ) is skew for i = 2, 3), and hence

θ4 = ε4, δ4 = γ(∈ Dτ ). Hence {ε1, ε4} is not contained in Dτ . This in turn


implies that {ε2, ε3} is not contained in Dτ (this is clear if ε2, ε3 are comparable -

then {ε2, ε3} = {ε1, ε4}; if (ε2, ε3) is skew, this follows from the fact that Dτ is a

sublattice of L).

Fact 6: (δ2, δ3) is skew

(Say, δ2 > δ3. We have

γ = δ4 = δ2 ∨ δ3 = δ2

which is not true, since (θ2, γ) is skew and δ2 = θ2 ∧ γ.)

In particular, we get the δ-diamond (⊆ Λi+1(Γ)).

Fact 7: If δ1 ∈ Dτ , then the δ-diamond is contained in Dτ (since δ4(= γ) ∈ Dτ ,

and Dτ is a embedded sublattice), and hence

Fθj≡ Fεj

(modM2τ ), j = 1, 2, 3, 4

If ε2, ε3 are comparable, say, ε2 > ε3, then ε2 = ε4, ε3 = ε1, and hence


τ )

It follows that fD ∈ b(τ) (cf. Lemma 5.1.1).

If (ε2, ε3) is skew, then by Lemma 5.1.3, we obtain that fD ≡ fD′(mod b(τ)), D′

being the ε-diamond (⊆ Λi+1(Γ)). For similar considerations, we may suppose

{δ2, δ3} is not contained in Dτ (since Dτ is a sublattice of L).

Hence we shall suppose that δ1 6∈ Dτ , and at least one of {δ2, δ3} is not in Dτ .

Fact 8: In view of Facts 5 & 7, we may suppose that none of the pairs

(ε1, ε4), (ε2, ε3), (δ1, δ4), (δ2, δ3) is contained in Dτ .

We now divide our discussion into the following two subcases:

Subsubcase B2(a1): At least one of {θ1, θ4} is in Eτ . Let us suppose that

θ1 ∈ Eτ (the proof for the case when θ4 ∈ Eτ being similar). Hence (by our


general reduction that a side of D is not contained in Eτ ), we obtain that θ2, θ3

are not in Eτ . This implies that for j = 2, 3 precisely one of {εj, δj} is in Dτ . From

this we get (in view of Fact 8) that precisely one of {ε2, δ3}, {ε3, δ2} is contained

in Dτ and the other has empty intersection with Dτ ; let us suppose that {ε2, δ3}is contained in Dτ , and ε3, δ2 6∈ Dτ (proof of the other case being similar). This

implies

Fθ2 ≡ Fδ2(modM2τ )

Fθ3 ≡ Fε3(modM2τ )

Fδ1 ≡ Fδ2(modM2τ )

(by considering the diamonds with skew diagonals (θ2, γ), (θ3, γ), (δ2, δ3) (cf. Fact

6 and the hypothesis that θ2, θ3 6∈ Λi+1(Γ)) respectively, and also observing that

ε2, δ3, δ4 are in Dτ ).

If ε1 ∈ Dτ , then Fθ1 ≡ Fδ1(modM2τ ), and hence we obtain

Fθ1 ≡ Fθ2(mod M2τ )

Hence it follows fD ∈ b(τ) (since Xθ1−Xθ2 , Xθ1 ∈ ker π(1), we get Xθ2 ∈ ker π(1)).

If ε1 6∈ Dτ , then ε3 ∈ Eτ (by consideration of the ε-diamond; note that (ε2, ε3) is

skew (for,ε2, ε3 comparable would imply {ε2, ε3} = {ε1, ε4} which is not possible

since ε2 ∈ Dτ while ε1, ε4(= θ4) are not in Dτ ). This together with the fact that

Fθ3 ≡ Fε3(modM2τ ) implies that Xθ3 ∈ ker π(1). Hence we obtain that fD ∈ b(τ)

(since by hypothesis, θ1 ∈ Eτ , and hence Xθ1 ∈ ker π(1)).

Subsubcase B2(a2): Both θ1, θ4 are not in Eτ .

This implies that ε1 ∈ Dτ (since δ1 6∈ Dτ (cf. Fact 7) ), and

(1) Fθ1 ≡ Fδ1(modM2τ )

(by considering the diamond with (θ1, γ) as the skew diagonal).


Further, precisely one of {ε2, ε3} is in Dτ ; this is clear if ε2, ε3 are comparable (in

which case {ε2, ε3} = {ε1, ε4}, and ε4(= θ4) 6∈ Dτ , and ε1 ∈ Dτ ). If (ε2, ε3) is skew,

then in the ε-diamond, we have that ε4(= θ4) 6∈ Eτ , and ε1 ∈ Dτ .

Let us then suppose that ε2 ∈ Dτ , ε3 6∈ Dτ (proof of the other case being similar).

Hence (by considering the diamond with (θ2, γ) as the skew diagonal) we obtain

(2) Fθ2 ≡ Fδ2(modM2τ )

Also,

(3) Fε3 ≡ Fθ4(modM2τ )

(If ε2, ε3 are comparable, then the facts that ε1, ε2 ∈ Dτ , ε4(= θ4) 6∈ Dτ imply that

ε1 = ε2, ε3 = θ4, and hence (3) follows. If (ε2, ε3) is skew, then by considering the

ε-diamond and using the facts that ε1, ε2 ∈ Dτ , we obtain that Fε3 ≡ Fε4(modM2τ ),

and (3) follows since ε4 = θ4.)

If δ3 ∈ Dτ , then by considering the δ-diamond (note (δ2, δ3) is skew (cf. Fact

6))) and the diamond with (θ3, γ) as skew diagonal, we obtain (noting that

δ4(= γ) ∈ Dτ )

(4) Fδ1 ≡ Fδ2(modM2τ )

(5) Fθ3 ≡ Fε3(modM2τ )

From (1),(2),(3),(4), (5) we obtain


τ )

Hence in view of Lemma 5.1.1, we obtain that fD ∈ b(τ).

If δ3 6∈ Dτ , then δ1 ∈ Eτ (note that in the δ-diamond, δ1 6∈ Dτ (cf. Fact 7),

δ2 6∈ Dτ (since ε2 ∈ Dτ ) while δ4 ∈ Dτ ); similarly, consideration of the diamond


with (θ3, γ) as the skew diagonal implies θ3 ∈ Eτ (note that ε3, δ3 6∈ Dτ ). The

facts that δ1, θ3 ∈ Eτ together with (1) imply that Xθ1 , Xθ3 are in ker π(1) and

hence fD ∈ b(τ).

Subcase B2(b): Both θ1, θ4 are in Λi+1(Γ) (and θ2, θ3 are not in Λi+1(Γ))

We have θ1 < γ < θ4 necessarily.

Fact 9: We have θ1 = δ1, θ4 = ε4, ε1 = γ = δ4.

As in B2(a) (cf. Fact 6), we have

Fact 10: (δ2, δ3), (ε2, ε3) are skew.

(The proof of the assertion that (ε2, ε3) is skew is similar to that of Fact 6; namely,

say ε2 > ε3. We have

γ = ε1 = ε2 ∧ ε3 = ε3

which is not true (since (θ3, γ) is skew, and ε3 = θ3 ∨ γ).

In particular, we get the δ-diamond and the ε-diamond (contained in Λi+1(Γ)).

Fact 11: In view of Facts 9 & 10, we have that none of the pairs

(ε1, ε4), (ε2, ε3), (δ1, δ4), (δ2, δ3) is contained in Dτ .

Subsubcase B2(b1): At least one of {θ1, θ4} is in Eτ .

Let us suppose that θ1 ∈ Eτ (the proof for the case when θ4 ∈ Eτ being similar).

Hence (by our general reduction that a side of D is not contained in Eτ ), we

obtain that θ2, θ3 are not in Eτ . This implies that for j = 2, 3 precisely one of

{εj, δj} is in Dτ . From this we get (in view of Fact 11) that precisely one of

{ε2, δ3}, {ε3, δ2} is contained in Dτ and the other has empty intersection with Dτ ;

let us suppose that {ε2, δ3} is contained in Dτ , and ε3, δ2 6∈ Dτ (proof of the other

5.2. Tangent cone at Pτ 72

case being similar). This implies

Fθ2 ≡ Fδ2(modM2τ )

Fδ1 ≡ Fδ2(modM2τ )

Fθ3 ≡ Fε3(modM2τ )

Fε3 ≡ Fε4(modM2τ )

(by considering the diamonds with skew diagonals (θ2, γ), (δ2, δ3), (θ3, γ), (ε2, ε3)

respectively). This combined with the fact that δ1 = θ1, ε4 = θ4 implies


τ )

Hence we obtain (in view of Lemma 5.1.1) that fD ∈ b(τ).

Subcase B2(b2): Both θ1, θ4 are not in Eτ .

This implies (in view of Facts 9 & 11) that precisely one of {ε2, ε3}, and that

precisely one of {δ2, δ3} is in Dτ (by considering the ε-diamond, δ-diamond

respectively); also, consideration of the diamond with (θj, γ), j = 2, 3 as the

skew diagonal implies that for j = 2, 3 at most one of {εj, δj} is in Dτ . Hence we

obtain that precisely one of {ε2, δ3}, {ε3, δ2} is contained in Dτ and the other has

empty intersection with Dτ . The rest of the proof is as in Subcase B2(b1), and

we obtain that fD ∈ b(τ).

This completes the proof of the case B2(b) and hence that of Proposition 5.1.7.

5.2 Tangent cone at Pτ

In this section, we construct “standard monomial basis” for TCPτ XL, the tangent

cone to XL at Pτ . We first determine J(τ). We then introduce a monomial order

> and determine in J(τ), the initial ideal of J(τ). Using MaCaulay’s Theorem,

we obtain a “standard monomial basis” for TCPτ XL.


Let X = SpecR ↪→ Al be an affine variety. Let P ∈ X, and let MP be the

maximal ideal in K[X] corresponding to P (we are concerned only with closed

points of X). Let A = OX,P , the stalk at P ; denote the unique maximal ideal in

A by M(= MP RMP). Then Spec gr(A,M), where gr(A,M) = ⊕

j∈Z+

M j/M j+1 is

the tangent cone to X at P , and is denoted TCP X. Note that

gr(R, MP ) = gr(A,M)

Remark 5.2.1. Let I(X) be the vanishing ideal of X for the embedding

X = Spec R ↪→ Al. Expanding f ∈ I(X) in terms of the local co-ordinates

at P , we have the following:

• TP (X), the tangent space to X at P is the zero locus of the linear forms of

f, f ∈ I(X).

• TCP (X), the tangent cone to X at P is the zero locus of the initial forms (i.e.,

form of smallest degree) of f, f ∈ I(X).

In the sequel, we shall denote the initial form of f by IF (f). We have a natural

surjective map

Sym(MP /M2P ) → gr(R, MP )

(here, Sym(MP /M2P ) is the symmetric algebra of MP /M2

P ). Thus we have a

natural embedding

TCP (X) ↪→ TP (X)(= Ar)

r being the dimension of TP (X), and TCP (X) is the zero locus of the ideal I in

Sym(MP /M2P ) such that gr(R, MP ) is the quotient Sym(MP /M2

P )/I (note that

I consists of IF (f), f ∈ I(X)).

We collect below some well-known facts on singularities of X:

Facts: 1. dimTP X ≥ dim X with equality if and only if X is smooth at P .


2. X is smooth at P if and only if multP X equals 1.

3. X is smooth at P if and only if the canonical surjective map Sym(MP /M2P ) →

gr(R, MP ) is an isomorphism, i.e., if and only if gr(R, MP ) is a polynomial

algebra.

5.2.2 Determination of I(τ):

We now take X = XL(= SpecK[Sσ]), and P = Pτ , τ being a face of σ (notation

as in §4.1). We shall denote XL by X, K[XL] by R. Let J(τ) be the kernel of

the surjective map

π : K[Xθ, θ ∈ L] → gr(R, Mτ ), Xθ 7→ Fθ

Note that J(τ) consists of the initial forms of elements of I(XL) (I(XL) being

the ideal in K[Xθ, θ ∈ L] generated by the diamond relations). For r ∈ N,

let π(r) be the restriction of π to the degree r part of the polynomial algebra

K[Xθ, θ ∈ L]; then ker π(r) = J(τ)r, the r-th graded piece of the homogeneous

ideal J(τ). By [15], we have that ker π(1) = ker g(1)τ ⊕ker h

(1)τ , ker h

(1)τ is generated

by {Xα, α ∈ Eτ} ∪ {(Xθ − Xδ), θ ∼ δ}; further, by Lemma 4.2.11, for any

θ ∈ L \ Dτ , there exists a µ ∈ Λτ (Γ) ∪ Eτ such that θ ∼ µ. In view of this

and Corollary 5.1.8, we may work with K[Xθ, θ ∈ Λτ (Γ)], and identify J(τ) with

the kernel of the surjective map

fτ : K[Xθ, θ ∈ Λτ (Γ)] → gr(R,Mτ ), Xθ 7→ Fθ

Note that J(τ) consists of the initial forms of elements of I(XΛτ (Γ)) (the ideal in

K[Xθ, θ ∈ Λτ (Γ)] generated by the diamond relations arising from diamonds in

Λτ (Γ)).


We shall next describe ker π(2), deduce a set of generators for ker π(r), r ∈ N, and

then determine a set of generators for J(τ). We first start with an analysis of the

diamond relations in Λτ (Γ).

For a diamond D in Λτ (Γ), with (θ, δ) as the skew diagonal, let

fD = XθXδ −Xθ∨δXθ∧δ

Certain quadratic relations on TCP XL: We have the following three types

of elements of J(τ)2 arising from the diamond relations:

(i) Let D be such that

• D ∩ Γ = ∅.

• One vertex, say α, is in Eτ (Eτ being as in §4.2). Denote the other vertex of

the diagonal through α by ξ.

• The diagonal not containing α has empty intersection with Eτ . Denote this

diagonal by (θ, δ).

We have that fD = XθXδ − XαXξ (in local co-ordinates around Pτ ) and hence

XθXδ −XαXξ belongs to J(τ)2. Now Xα ∈ J(τ)1 (cf. Lemma 4.2.6). Hence we

obtain that XθXδ is in J(τ)2.

(ii) Let D be such that D ∩ {Γ ∪ Eτ} = ∅. Then fD remains homogeneous (in

local co-ordinates around Pτ ) and belongs to J(τ)2.

(iii) Let D1, · · · , Dr be a set of diamonds such that there exists a pair (α, γ) in

Λτ (Γ), α ∈ Eτ , γ ∈ Γ such that (α, γ) is a diagonal of Di; note that (α, γ) is

necessarily the main diagonal of Di (since α ∈ Λτ (Γ)), and the other diagonal

of Di has empty intersection with Γ (since any γ ∈ Γ can not be on the skew

diagonal of any diamond in Λτ (Γ)). Further, for each i, let the other diagonal of

Di have empty intersection with Eτ . Denote this diagonal by (θi, δi); note that


by Lemma 4.2.13, if r > 1, then γ is uniquely determined by α. If r > 1, let us

further suppose that D1, · · · , Dr are all the diamonds in Λτ (Γ) having (α, γ) as

the main diagonal.

(a) If r = 1 (i.e., there exists a unique skew pair (θ, δ) with {θ∨δ, θ∨δ} = {α, γ}),then denoting the corresponding diamond by D, we have IN(fD) = Xα (and

Xα ∈ J(τ)1).

(b) If r > 1, then for a pair (i, j), i 6= j, we have, IF (fDi− fDj

) ≡ XθiXδi

−Xθj

Xδj(modJ(τ)); hence we obtain that Xθi

Xδi−Xθj

Xδj∈ J(τ)2.

In the sequel, we shall refer to a D as in (i) (resp.(ii),(iii)) as a Type I (resp. Type

II,Type III ) diamond. The elements of J(τ) appearing in (i)(resp. (ii),(iii)(a),(b))

above will be referred to as a Type I-element (resp. Type II,III-elements) of J(τ).

Ideal generators for J(τ):

Let a(τ) denote the ideal in the polynomial algebra K[Xθ, θ ∈ Λτ (Γ)] generated

by elements in J(τ) of Type I,II,III (as described above). Clearly a(τ) ⊆ J(τ).

Theorem 5.2.3. The inclusion a(τ) ⊆ J(τ) is an equality.

Proof 5.2.4. Let h ∈ I(XΛτ (Γ)). We are required to show that IF (h) belongs to

a(τ). We shall first write h as a polynomial expression in the fD’s, and then find

IF (h). Now h is a polynomial expression in the fD’s, D being of Type I,II or III

of §5.2.2. Let

h = H1 + H2 + H3

where H1, H2, H3 are polynomial expression in the fD’s D’s being of Type I,II ,

III respectively. Let d be the degree of IF (h). Now for any l, the degree l forms

of H1, H2 are clearly in a(τ) (since for D of Type I,II, fD is in a(τ)). Thus we are


led to analyze the degree l forms in H3 (for any l). Let

(∗) H3 =∑

α∈{Λτ (Γ)∩Eτ}

∑i

Aα,i[Xα(1−Xγ)−XθiXδi

]

where (α, γ), (θi, δi) are the diagonals of diamonds of Type III. Since Xα ∈ a(τ)

for α ∈ Eτ , we are led to analyze the degree d forms in a H3 as above, where we

may suppose that none of the monomials present in Aα,i’s is divisible by any Xα′

for α′ ∈ Eτ . If∑


∑i

Aα,iXα 6= 0

then the result is clear (since, Xα ∈ a(τ), for α ∈ Eτ ). Let then

∑


∑i

Aα,iXα = 0

Hence we obtain

(∗∗)∑

i

Aα,i = 0, ∀α

(in view of our assumption that none of the monomials in Aα,i’s is divisible by

any Xα′ for α′ ∈ Eτ ). Hence we obtain

H3 = −∑


∑i

Aα,i XθiXδi

Now (**) implies that if a certain monomial m occurs with a coefficient s in∑i

Aα,i, then −m also occurs with the same coefficient s in∑i

Aα,i. Hence

denoting by Mα the set of all monomials appearing in the polynomial∑

i Aα,i,

we have,

(∗ ∗ ∗) H3 =∑

α

∑

m∈Mα

m(XθiXδi

−XθjXδj

)

where (θi, δi), (θj, δj) are skew diagonals of diamonds Di, Dj in Λτ (Γ) for which

(α, γ) is the main diagonal (cf. (iii) of §5.2.2). The required result now follows


(note that (***) implies that H3 is in the homogeneous ideal generated by Type

III (b) elements of §5.2.2, and hence any homogeneous component - in particular,

IN H3 is again in that ideal).

A monomial order in K[Xθ, θ ∈ Λτ (Γ)]: Take a total order Â on Λτ (Γ)

extending the partial order. Define Xθ Â Xδ if θ ≺ δ; a monomial will be

written as Xθ1 · · ·Xθr where θ1 º · · · º θr. Given two monomials m1,m2, define

m1 Â m2 if either deg m1 > deg m2 or deg m1 = deg m2, and m1 ≥ m2 (under

the lexicographic order induced by Â, i.e., if m1 = Xθ1 · · ·Xθr ,m2 = Xδ1 · · ·Xδr ,

then there exists a t < r such that θi = δi, i < t and θt Â δt).

Let in J(τ) be the initial ideal of J(τ); note that

in J(τ) = {in f, f ∈ J(τ)}

where for a polynomial g ∈ K[Xθ, θ ∈ L], in g is the greatest (under Â) monomial

occurring in g. Let

Eτ (Γ) = Λτ (Γ) ∩ Eτ

Let notation be as in §5.2.2. Further, given an equivalence class [θ] we shall

denote the maximal element (under Â) in {δ, δ ∼ θ} by [θ]max

Proposition 5.2.5. in J(τ) is generated by

{Xθ, θ 6= [θ]max},

{Xα, α ∈ Eτ (Γ)},

{XθXδ, (θ, δ) being a diagonal of a diamond of Type I}

{XθXδ, (θ, δ) being the skew diagonal of a diamond of Type II}

{XθXδ, (θ, δ) being the skew diagonal of a diamond of Type III with r > 1,

and (θ, δ) 6= max {([θi]max, [δi]max)} }


(Note that in the last set (θi, δi) are all the skew diagonals such that

θi ∨ δi = θ ∨ δ, θi ∧ δi = θ ∧ δ with the main diagonal as in Type III.)

Proof 5.2.6. Let bτ be the ideal generated by the set of monomials as described

in the Proposition; clearly, we have that bτ ⊆ in J(τ). We have that in J(τ)

consists of initial monomials in the initial forms of elements of I(XΛτ (Γ)) (the

ideal in K[Xθ, θ ∈ Λτ (Γ)] generated by diamond relations arising from diamonds

in Λτ (Γ)). Now a typical element h ∈ I(XΛτ (Γ)) is a polynomial expression in the

fD’s, D being of Type I,II or III of §5.2.2. Let

h = H1 + H2 + H3

where H1, H2, H3 are polynomial expression in the fD’s D’s being of Type I,II ,

III respectively. Proceeding as in the proof of Proposition 5.2.3, we see that in h

is in bτ .

Definition 5.2.7. Call a monomial in K[Xθ, θ ∈ Λτ (Γ)] standard if it is not in

the collection as described in the above Proposition.

As an immediate consequence of MaCaulay’s theorem (cf. [6], Theorem 15.3), we

have the following

Theorem 5.2.8. The standard monomials form a basis for the ring of regular

functions on TCpτ X(L).

Theorem 5.2.9. Let τ be such that Eτ (Γ) is empty. Further, for each θ ∈ L, let

[θ] consist of just {θ}. Then TCpτ X(L) is again a Hibi variety.

Proof 5.2.10. Hypothesis implies that J(τ) is generated by just Type II

elements, namely, the diamond relations in Λτ (Γ). Thus we obtain that

TCpτ X(L) is simply the Hibi variety associated to the distributive lattice Λτ (Γ).

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