vanishing products of one-forms and critical points of master...

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Advanced Studies in Pure Mathematics 62, 2012Arrangements of Hyperplanes—Sapporo 2009pp. 75–107

Vanishing products of one-formsand critical points of master functions

in memory of V. I. Arnol’d

Daniel C. Cohen1, Graham Denham2, Michael Falk andAlexander Varchenko3

Abstract.

Let A be an a!ne hyperplane arrangement in C! with complementU . Let f1, . . . , fn be linear polynomials defining the hyperplanes of A,and A· the algebra of di"erential forms generated by the one-formsd log f1, . . . , d log fn. To each ! ! Cn we associate the master function# =

!ni=1 f

"ii on U and the closed logarithmic one-form " = d log#.

We assume " is a general element of a rational linear subspace D ofA1 of dimension q > 1 such that the map

"k(D) " Ak given bymultiplication in A· is zero for all p < k # q, and is nonzero fork = p. With this assumption, we prove the critical locus crit(#) of #has components of codimension at most p, and these are intersectionsof level sets of p rational master functions. We give conditions thatguarantee crit(#) is nonempty and every component has codimensionequal to p, in terms of syzygies among polynomial master functions.

If A is p-generic, then D is contained in the degree p resonancevariety Rp(A)—in this sense the present work complements previouswork on resonance and critical loci of master functions. Any arrange-ment is 1-generic; we give a precise description of crit(#") in case ! liesin an isotropic subspaceD of A1, using the multinet structure on A cor-responding to D $ R1(A). This is carried out in detail for the Hessianarrangement. Finally, for arbitrary p and A, we establish necessaryand su!cient conditions for a set of integral one-forms to span such asubspace, in terms of nested sets of A, using tropical implicitization.

Received October 17, 2010.Revised March 27, 2011.2010 Mathematics Subject Classification. Primary 32S22; Secondary 55N25,

52C35, 14T04.Key words and phrases. Arrangement of hyperplanes, master function, crit-

ical locus, resonance variety, Orlik–Solomon algebra, tropicalization.1Partially supported by National Security Agency grant H98230-05-1-0055.2Partially supported by a grant from NSERC of Canada.3Partially supported by NSF grant DMS-0555327.

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76 D. C. Cohen, G. Denham, M. Falk and A. Varchenko

§1. Introduction

Let A = {H1, . . . , Hn} be an arrangement of distinct a!ne hyper-planes in C!, with complement U = C! %

#ni=1 Hi. Choose a linear

polynomial fi with zero locus Hi, for each i. Let ! = (!1 . . . ,!n) ! Cn,and consider the master function

#" =n$

i=1

f"ii .

The multi-valued function #" has a well-defined critical locus

crit(#") = {x ! U | d#"(x) = 0}.

Indeed, crit(#") coincides with the zero locus V ("") of the single-valuedclosed logarithmic one-form

"" = d log(#") =n%

i=1

!id log(fi).

In particular, crit(#") is unchanged if ! is multiplied by a non-zeroscalar. We are interested in the relation between crit(#") and algebraicproperties of the cohomology class represented by "" in H1(U,C).

For certain arrangements A and weights !, the critical points of #"

yield a complete system of eigenfunctions for the commuting hamilito-nians of the sln(C)-Gaudin model, via the Bethe Ansatz [23, 25, 17].That application was the origin of the term master function, introducedin [28]. Much of that theory depends only on combinatorial properties ofarrangements, and can be formulated in that general setting—see [29].

Let A· denote the graded C-algebra of holomorphic di"erential formson U generated by {d log(fi) | 1 # i # n}. By a well-known result ofBrieskorn, the inclusion of A· into the de Rham complex of U inducesan isomorphism in cohomology, and thus A· &= H ·(U,C), see [1, 4]. Inparticular A1 &= Cn. Since "" ' "" = 0, left-multiplication by "" makesA· into a cochain complex (A·,""). For generic !, Hp(A·,"") = 0for p < #, and dimH!(A·,"") = |$(U)|, see [24, 30]. At the sametime, for generic !, #" has |$(U)| isolated, nondegenerate critical points[27, 20, 26]. Here we are concerned with Hp(A·,"") and crit(#") when! is not generic.

For p < #, the " for which Hp(A·,") (= 0 comprise the pth resonancevariety Rp(A) of A, a well-studied invariant of A·. On the other hand,precise conditions on ! guaranteeing that #" has |$(U)| isolated criticalpoints are not known. There are also examples where the critical pointsof #" are isolated but degenerate.

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Vanishing products 77

In some cases crit(#") is positive-dimensional. If A is a discrim-inantal arrangement, in the sense of [24], then for certain choices ofintegral weights ! arising from a simple Lie algebra g, crit(#") has com-ponents of the same positive dimension [25, 16, 18]. In the particularcase g = sl2(C) of this situation, the codimension of crit(#") is #% 1. Inthis case, it was shown in [6] that "" ! R!!1(A) for these !, with therank of the skew-symmetric part of H!!1(A·,"") equal to the numberof components of crit(#").

Our work in [5] provides a weak generalization of these results.There we study the universal critical set, the set $ of pairs (x, a) suchthat x ! V ("a). For fixed !, crit(#") is the a = ! slice of $. Let $ bethe Zariski closure of $ in C! ) Cn, and $" the a = ! slice of $. In [5]we show, if "" ! Rp(A), then $" has codimension at most p, providedA is tame and either p # 2 or A is free. See [5] for definitions of free andtame arrangements; any a!ne arrangement in C2 is tame. It is not truein general that $" is the closure of $". Indeed, $" may be empty underthe given hypotheses—that is, $" $ C! ) Cn may lie over

#ni=1 Hi.

In this paper, we obtain somewhat more precise information oncrit(#"), for more general arrangements, but impose a di"erent hypoth-esis on "". Namely, we assume that "" has a decomposable cocycle, thatis, there exists % ! Ap such that "" ' % = 0, and % is a product of pelements of A1, whose linear span does not include "".

We say a subspace D of A1 is singular if the multiplication map"q(D) " Aq is zero, where q = dimD. Let p be maximal such that"p(D) " Ap is not the zero map. If " = d log(#) ! D%{0}, then " canbe included in a basis of D, and any p-fold product of the other basiselements is a decomposable p-cocycle for ".

We assume that D is a rational subspace of A1, that is, D has abasis % = {"#1 , . . . ,"#q} with each "#j an integer linear combinationof d log(f1), . . . , d log(fn). Associated to % is a rational mapping #! =(##1 , . . . ,##q ) : C! ! Cq, whose image is a quasi-a!ne subvariety Y =Y! of Cq. The dimension of Y is p. If " = d log(#) ! D, then thecritical locus crit(#) is consists of fibers of #! and singular points of#!. In particular, for generic " ! D, crit(#) has codimension at mostdim(Y ) = p. We obtain more precise conclusions in case the projectiveclosure Y is a curve (p = 1) or a hypersurface (p = q % 1), or Y isnonsingular and meets the coordinate hyperplanes transversely. If Y islinear, of any codimension, with Y = Y * (C")q and #! nonsingular,we get a complete description of the crit(#) for d log(#) ! D, in termsof critical loci of master functions on the complement of the rank-parrangement cut out on Y by the coordinate hyperplanes.

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Every component of R1(A) is a rationally-defined and isotropic lin-ear subspace [14], and every element of R1(A) has a decomposable co-cycle. Moreover, by the theory of multinets and Ceva pencils [10], wecan choose % so that the variety Y! corresponding to a component ofR1(A) is linear. We carry out the entire analysis in detail in this case,with special attention to the Hessian arrangement, the one case we knowfor which Y! is not a hypersurface.

Our approach lends itself to tropicalization, using the main result of[7]. Using the nested set subdivison of the Bergman fan [11], we derivea rank condition for a product of integral one-forms "#1 ' · · · ' "#q tovanish. The rank condition can be used in case A is p-generic to givea combinatorial description of the (p+ 1)-tuples of integral forms in A1

whose product vanishes, analogous to the description of R1(A) in termsof neighborly partitions—see [2].

The outline of this paper is as follows. In Section 2 we introduceOrlik–Solomon algebras and resonance varieties, prove a general resultabout zero loci of di"erential forms, and compute critical loci directlyfor some examples, including the Hessian arrangement. In Section 3 weconsider logarithmic one-forms with decomposable p-cocycles satisfyingthe rationality criterion above, obtaining a precise description of theirzero loci, especially in case #! is nonsingular and Y ! is a hypersurfacemeeting the coordinate hyperplanes transversely. We revisit the exam-ples from Section 2. In Section 4 we treat the case where Y ! is linear,returning to the example of the Hessian arrangement. In Section 5 weformulate a test for existence of decomposable cocycles using tropicalimplicitization.

§2. Resonance, vanishing products, and zeros of one-forms

It will be more convenient for us to consider arrangements of pro-jective hyperplanes in complex projective space P!. Let [x0 : · · · : x!]be homogeneous coordinates on P!, and let &i : C!+1 " C be a nonzerohomogeneous linear form, for 0 # i # n. Assume without loss that&0(x) = x0. Let Hi = ker(&i), considered as a projective hyper-plane in P!, and let A = {H0, . . . , Hn}. We will denote the corre-sponding linear hyperplanes in C!+1 by cHi, comprising the central ar-rangement cA = {cH0, . . . , cHn}. Let U = P! %

#ni=0 Hi. We identify

[1 : x1 : · · · : x!] ! P! %H0 with (x1, . . . , x!) ! C!, and set

fi(x1, . . . , x!) = &i(1, x1, . . . , x!)

for 1 # i # n. Then we recover the a!ne arrangement A of the Intro-duction, with the same complement U . In P!, U is the complement of

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the singular projective hypersurface defined by

Q =n$

i=0

&i,

the (homogeneous) defining polynomial of A.

2.1. The projective Orlik–Solomon algebra

Let &·(U) be the complex of holomorphic di"erential forms on U .The Orlik–Solomon algebra of A is the subalgebra A·(A) of &·(U) gen-erated by d log(fi), 1 # i # n, as in the Introduction.

We will also study A·(A) in homogeneous coordinates. Let "i =d log(&i) for 0 # i # n, and let A·(cA) be the algebra of holomorphicforms on C!+1 generated by "0, . . . ,"n. Define ' : A·(cA) " A·(cA) by

'("i1 ' · · · ' "ik) =k%

j=1

(%1)j!1"i1 ' · · · ' &"ij ' · · · ' "ik

and extending linearly. Then ' is a graded derivation of degree -1, and

'' n%

i=0

ci"i

(=

n%

i=0

ci.

In general, a holomorphic p-form on C!+1 % {0} descends to a well-defined form on P! if and only if it is C"-invariant and its contractionalong the Euler vector field

)ni=0 xi

$$xi

vanishes, see [8]. This contrac-tion, on A·(cA), is given by ', and A·(cA) consists of C"-invariant forms.Then we may identify A·(A) with the subalgebra ker(') of A·(cA). Thisis easily seen to coincide with the subalgebra of A·(cA) generated byker(') *A1(cA).

With our choice of coordinates, d log(fi) = "i % "0 under this iden-tification. {"1 % "0, . . . ,"n % "0} generates A· by the remark above.Also, (A·(cA), ') is an exact complex, so that im(') = ker(') = A·(A),see [19].

There is a well-known presentation of A·(cA) as a quotient of theexterior algebra E· =

"·(e0, . . . , en). For C = {i1, . . . , ik} $ {0, . . . , n},write eC = ei1 · · · eik ! Ek. Say C is a circuit of cA if C is minimal withthe property that

codim*

j#C

Hj < |C|.

Then A·(cA) is isomorphic to E·/I, where

I =''eC | C is a circuit of cA

(.

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2.2. Resonance varieties

Let " =)n

i=0 !i"i, where ! = (!0, . . . ,!n) ! Cn+1. Assume that'" =

)mi=0 !i = 0. Since d log(fi) = "i % "0, " =

)ni=1 !i d log(fi).

Then " ! A1, and " ' " = 0, so we obtain a cochain complex

0 %" A0 %$!%%%" · · · %$!%%%" Ap %$!%%%" · · · %$!%%%" A! %" 0.

Let

Zp(") = {% ! Ap | " ' % = 0},Bp(") = {% ! Ap | % = " ' ( for some ( ! Ap!1}, and

Hp(A·,") = Zp(")/Bp(").

ThenRp(A) = {" ! A1 | Hp(A·,") (= 0}

is, by definition, the pth resonance variety of A.As observed in the Introduction,

" = d log(#) =d#

#,

where # =!n

j=1 f"j

j , and crit(#) coincides with the zero locus of ". In

homogeneous coordinates, # is given by!n

j=0 &"j

j .

2.3. Zeros of forms

We start with an elementary observation about products and zerosof di"erential forms. If % ! &k(U), for some k, 0 # k # #, let

V (%) = {x ! U | %(x) = 0},

a quasi-a!ne subvariety of C!. Let U(%) = U % V (%).

Proposition 2.1. Suppose " ! &1(U) and % ! &p(U) satisfy " '% = 0. Then every component of V (") % V (%) has codimension lessthan or equal to p.

Proof. We may write " =)!

i=1 bidxi for some holomorphic func-tions b1, . . . b! on U . Then

V (") =!*

i=1

V (bi).

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Similarly, % =)

I AIdxI for some holomorphic functions AI , whereI ranges over all subsets I = {i1, . . . , ip}< of {1, 2, . . . , #}, and dxI =dxi1 ' · · · ' dxip . (The subscript “<” is meant to indicate that theelements of I are listed in increasing order.) Set UI = U(AI), and letSI denote the coordinate ring of UI , i.e., SI is C[x1, . . . , x!], localized atAI . Then

U(%) =+

I

UI .

The equation " ' % = 0 says, for each subset J of {1, . . . , #} of sizep+ 1,

(2.1)%

i#J

)(i, J)biAJ!{i} = 0.

Here )(i, J) = ±1 depending on the position of i in J .We have

V (") * U(%) =+

I

V (") * UI .

Fix I = {i1, · · · , ip}<. For each i (! I, set J = I + {i} in equation(2.1). Since AI (= 0 on UI , one can solve for bi in terms of bi1 , . . . , bip .This means bi lies in the ideal (bi1 , . . . , bip) of SI . Since this holdsfor every i (! I, the defining ideal of V (") * UI in SI is contained in(bi1 , . . . , bip). Then each irreducible component of V (") * UI has codi-mension less than or equal to the codimension of (bi1 , . . . , bip), which isat most p. Since the UI cover U(%), the result follows. Q.E.D.

Corollary 2.2. If*

{V (%) | % ! &p(U)," ' % = 0} = ,,

then every component V (") has codimension less than or equal to p.

Corollary 2.3. Suppose X is a component of V (") of codimensionc. If % is a p-form satisfying " ' % = 0 and p < c, then X $ V (%).

Remark 2.4. The preceding results go through without change forany smooth complex analytic variety U , interpreting x1, . . . , x! as localholomorphic coordinates on U .

A p-form % satisfying " '% = 0 will be called a p-cocycle for ". Wesay % is trivial if % = " ' ( for some ( ! &p!1(U). If % is a trivialcocycle for ", then V (") $ V (%).

The trivial cocycle condition % = " ' ( is generally di!cult tocharacterize. We propose the following conjecture, the converse to theobservation above.

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Conjecture 2.5. If % ! Ap, then % = " ' ( for some ( ! Ap!1 ifand only if " ' % = 0 and V (") $ V (%).

The conjecture is not hard to prove directly in case p = 1, and thestatement for any p follows from results of [29] if " ! A1 is generic.

2.4. Examples

Our first example is linearly equivalent to the rank-three braid ar-rangement.

Example 2.6. Let A = {H0, . . . , H5} be the arrangement withdefining polynomial

Q = xyz(x% y)(x% z)(y % z),

with the hyperplanes labelled according to the order of factors in Q.For a, b, c ! C, not all zero, with a+ b+ c = 0, let

#abc = [x(y % z)]a[y(x% z)]b[z(x% y)]c.

Then "abc := d log(#abc) = a("0 + "5) + b("1 + "4) + c("2 + "3).One computes

"abc = [dx dy dz]

,

-b1b2b3

.

/

= [dx dy dz]

,

-1/x 1/(x% z) 1/(x% y)

1/(y % z) 1/y 1/(y % x)1/(z % y) 1/(z % x) 1/z

.

/

,

-abc

.

/ .

The zero locus V ("abc) is given by the vanishing of b1, b2, and b3. Thekernel of the matrix is spanned by (x(y % z), y(z % x), z(x % y)), so[x : y : z] ! V ("abc) if and only if [x(y%z) : y(z%x) : z(x%y)] = [a : b : c].Since a + b + c = 0, this is equivalent to [x(y % z) : y(z % x)] = [a : b],i.e.,

x(y % z)

y(z % x)=

a

b

or, more symmetrically,

a

x+

b

y+

c

z= 0.

If any of a, b, or c are zero, then crit(#abc) = V ("abc) is empty. Otherwisecrit(#abc) has codimension 1. Moreover, crit(#") is a level set of the

master function x(y!z)y(z!x) .

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Fig. 1. The braid arrangement with crit(#1,1,!2)

Let D = {"abc | a+ b+ c = 0}. Then D $ A1 and, for any " ! D,Z1(") = D. Thus % ! Z1("abc) if and only if % = d log#a!b!c! witha% + b% + c% = 0. Then % has zero locus given by

a%

x+

b%

y+

c%

z= 0,

which one can see is disjoint from V ("abc) if and only if % (! B1("abc).The Zariski closure of every nonempty critical set contains the fourpoints [0 : 0 : 1], [1 : 1 : 1], [1 : 0 : 0], and [0 : 1 : 0]—see Figure 1.

Here is a rank-four example.

Example 2.7. Let A = {H0, . . . , H7} be the arrangement of eightplanes in P3 with defining polynomial

Q = xyzw(x+ y + z)(x+ y + w)(x+ z + w)(y + z + w).

The dual point configuration consists of the four vertices and four face-centers of the 3-simplex. Fix a, b, c, d ! C and let

# =

0x

y + z + w

1a0 y

x+ z + w

1b0 z

x+ y + w

1c0 w

x+ y + z

1d

.

The one-form " = d log# belongs to a 4-dimensional component ofR2(A). If none of a, b, c, d are zero, then H1(A,") = 0 and H2(A,") &=C. A nontrivial 2-cocycle for " is given by

% = b · '("167) + c · '("257) + d · '("347).

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One sees that " is equal to the product

[dx dy dz dw]

,

222-

1x

!1x+z+w

!1x+y+w

!1x+y+z

!1y+z+w

1y

!1x+y+w

!1x+y+z

!1y+z+w

!1x+z+w

1z

!1x+y+z

!1y+z+w

!1x+z+w

!1x+y+w

1w

.

333/

,

22-

abcd

.

33/ .

Computing the kernel of the matrix, we see that [x : y : z : w] ! V (") ifand only if the vector (x(y+z+w), y(x+z+w), z(x+y+w), w(x+y+z))is proportional to (a, b, c, d), giving the equations

x(y + z + w)

w(x+ y + z)=

a

d,

y(x+ z + w)

w(x+ y + z)=

b

d,

z(x+ y + w)

w(x+ y + z)=

c

d.

V (") has a component of codimension two, given by

x+ y + z + w = 0,a

x+

b

y+

c

z+

d

w= 0.

For general a, b, c, d, the remaining components of V (") consist of fouradditional points in P3. It follows from Corollary 2.3 that the cocycle %vanishes on the isolated points of V ("). This can also be verified hereby direct computation.

Example 2.8. The Hessian arrangement consists of the 12 linesthrough the inflection points of a nonsingular cubic in P2. It is the onlyknown arrangement of rank greater than two that supports a globalcomponent of R1(A) of dimension greater than two. That is, there isan element " ! A1 which has poles along every hyperplane of A, andsatisfies dimH1(A·,") > 1.

Any nonsingular cubic is equivalent to

(2.2) x3 + y3 + z3 % 3txyz = 0

up to projective transformation, for some t ! C. These cubics havethe same inflection points. Then, up to projective transformation, theHessian arrangement A is defined by

Q = xyz(x+ y+ z)(x+ y+ *z)(x+ y+ *2z)(x+ *y+ z)(x+ *y+ *z)

· (x+ *y + *2z)(x+ *2y + z)(x+ *2y + *z)(x+ *2y + *2z),

where * = e2!i3 . The 12 lines of A are the irreducible components of the

four singular cubics in the family 2.2, corresponding to t = -, 1, *, and*2. See [19, Example 6.30].

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Numbering the hyperplanes in order, these singular cubics are givenby

P0 = &0&1&2 = xyz,

P1 = &3&8&10 = x3 + y3 + z3 % 3xyz,

P2 = &4&6&11 = x3 + y3 + z3 % 3*xyz, and

P3 = &5&7&9 = x3 + y3 + z3 % 3*2xyz.

For (a1, a2, a3) ! C3 let " = "a1a2a3 = d log(#), where

# = #a1a2a3 =

0P1

P0

1a10P2

P0

1a20P3

P0

1a3

.

Let D $ A1 be the space of all such forms. Then H1(A·,") &= D/C"has dimension two.

As in the previous examples, we write

" =4dx, dy, dz

5M

,

-a1a2a3

.

/ ,

where M is a 3) 3 matrix of rational functions, the Jacobian of

(log(P1/P0), log(P2/P0), log(P3/P0)).

Then "(x) = 0 for x ! U if and only if a = (a1, a2, a3) lies in the kernelof M(x).

The matrix M has rank 1. In fact one finds that M = vwT where

v =

,

2-

2x3!y3!z3

xx3!2y3+z3

yx3+y3!2z3

z

.

3/ and w =

,

-1P11P21P3

.

/ .

The critical equation becomes vwT a = 0, which is satisfied if andonly if wT a = 0 or all components of v vanish. The latter occurs at thepoints given by x3 = y3 = z3, which are the common inflection points ofthe cubics (2.2). In particular those points do not lie in the complementU . Thus crit(#) is defined by the single equation

(2.3)a1P1

+a2P2

+a3P3

= 0.

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Then crit(#) is empty or has codimension one and degree six. Work-ing in the torus xyz (= 0, set

T =x3 + y3 + z3

xyz.

Then equation (2.3) is equivalent to

(2.4)a1

T % 3+

a2T % 3*

+a3

T % 3*2= 0,

which becomes a quadratic AT 2 +BT + C = 0 in T .Then, if a = (a1, a2, a3) is generic, crit(#) has two irreducible com-

ponents. Each component is the intersection with U of a level set T = 3tof T . These are nonsingular fibers in the pencil (2.2), meeting eachother and A at their nine common inflection points. So crit(#) $ U hastwo connected components. Every pair of nonsingular fibers appears ascrit(#) for some a. The discriminant B2%4AC defines a hypersurface in(a1, a2, a3)-space for which the corresponding critical locus crit(#) has asingle nonreduced component, and every nonsingular fiber can appear.

For some values of a, crit(#) is empty or has only one reducedcomponent. This is easiest to see by clearing fractions in (2.3), to obtain

(2.5) a1P2P3 + a2P1P3 + a3P1P2 = 0.

When one component of the variety defined by (2.5) is Pi = 0 or P0 =xyz = 0, then crit(#) has one reduced component. This occurs if a isa generic point on ai = 0 or a1 + a2 + a3 = 0. If ai = 0 for some iand a1 + a2 + a3 = 0, or if ai (= 0 for only one i, then (2.5) becomesPiPj = 0, and crit(#) = ,. If a is a cyclic permutation of (0, *, 1) orequals (1, *, *2), up to scalar multiple, then (2.5) becomes P 2

i = 0, andcrit(#) = ,.

Each cocycle of " = "a1a2a3 has the form % = "b1b2b3 for someb1, b2, b3. So we see that V (%) and V (") can have a component incommon, but if " and % are not proportional, then V (") % V (%) isnonempty and has codimension one.

§3. Decomposable cocycles

We will now assume % is a cocycle for " ! A1 and % is a product oflogarithmic one-forms. Then " is a factor in a vanishing product of p+1one-forms in A·. To carry out our geometric analysis it is necessary towork initially over the integers, although eventually the results extend toC-linear combinations of the original integral weights. For that reasonwe state the result in terms of subspaces of A1.

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We will be dealing with rational functions parametrizing a!ne orprojective algebraic varieties. For that reason we formulate and proveour results algebraically. For any quasi-projective variety X, write C[X]for the ring of regular functions on X, and C(X) for the field of rationalfunctions on X. If X is a subvariety of projective space, then elementsof C[X] are represented by homogeneous polynomials and elements ofC(X) by homogeneous rational functions of degree zero. If ( : R " S isa homomorphism of C-algebras, denote by &S|R the S-module of Kahlerdi"erentials of S over R. Write &[X] = &C[X]|C and &(X) = &C(X)|C.Elements of &[X] (resp. &(X)) are polynomial (resp. rational) one-forms on X.

3.1. Singular subspaces of A1

Let D be a subspace of A1. We call D a singular subspace if themultiplication map

"q(D) " Aq is the zero map, where q = dimD. Therank of D is the largest p such that

"p(D) " Ap is not trivial. We say Dis rational if D has a basis {"#1 , . . . ,"#q}, with +i = (+i0, . . . , +in) ! Zn+1

for 1 # i # q. Then ##i =!n

j=1 f#ijj =

!nj=0 &

#ijj is a single-valued

rational function on P!, regular on U . (Recall)n

j=0 +ij = 0.)We apply the following general result. It is an easy consequence of

the implicit function theorem, but we give an algebraic proof that holdsover any algebraically-closed field of characteristic zero. See [13] and [9]for the relevant background on Kahler di"erentials.

Proposition 3.1. Suppose F1, . . . , Fq are rational functions on C!,and

F = (F1, . . . , Fq) : C! ! Cq.

Then the image of F has dimension less than k if and only if

dFi1 ' · · · ' dFik = 0

for all 1 # i1 < · · · < ik # q.

Proof. The image of F is a quasi-a!ne variety, whose functionfield is isomorphic to C(F1, . . . , Fq). Then the dimension p of im(F )is equal to the transcendence degree of C(F1, . . . , Fq) over C. With-out loss of generality, suppose {F1, . . . , Fp} is a transcendence base forC(F1, . . . , Fq) over C. Then the set {dF1, . . . , dFp} forms a basis for&(C(F1, . . . , Fq)), a vector space over C(F1, . . . , Fq), see [9, Theorem16.14]. Then

dF1 ' · · · ' dFp (= 0.

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If {Fi1 , . . . , Fik} $ {F1, . . . , Fq} with k > p, then {dFi1 , . . . , dFik} islinearly dependent over C(F1, . . . , Fq), hence

dFi1 ' · · · ' dFik = 0.

Q.E.D.

If % = (+1, . . . , +q) with +i ! Zn+1 satisfying)n

j=0 +ij = 0, set

#! := (##1 , . . . ,##q ) : P! ! Cq.

Proposition 3.1 applies as follows.

Corollary 3.2. Suppose D is a rational singular subspace of A1,and #! is the rational mapping associated to an ordered integral basis %of D. Then dim#!(U) is equal to the rank of D. In particular, #!(U)has positive codimension in Cq.

Let [z0 : · · · : zq] be homogeneous coordinates on Pq, and identifyCq with the Pq % {z0 = 0} as above. Let Y be the Zariski closure ofY = #!(U) in Pq. In homogeneous coordinates, #! is given by

#! = [1 : ##1 : · · · : ##q ] : P! ! Pq.

Clearing fractions, we have

#! = [#&0 : #&1 : · · · : #&q ]

where the master functions #&i are homogeneous polynomials of thesame degree d. Moreover we may assume the #&i have no commonfactors. Equivalently, the weights ,i = (,i0, . . . , ,in) are non-negativeinteger vectors with

)ni=0 ,ij = d for 0 # i # q, whose supports have

empty intersection. Then

##i =#&i

#&0

, +i = ,i % ,0, and "#i = "&i % "&0 .

Definition 3.3. % is essential if every hyperplane H ! A appearsas a component of V (#&i) for some i, 0 # i # q.

Henceforth we will tacitly assume % is essential. We have Y $Y * (C")q in any case. If % is not essential, the inclusion is proper, andmay be proper otherwise—see Example 3.16.

The defining ideal I = I! of Y is generated by homogeneous polyno-mials P (z0, . . . , zq) for which P (#&0 , . . . ,#&q ) vanishes identically on U ,or, equivalently, P (1,##1 , . . . ,##q ) = 0. We will sometimes refer to I!

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as the syzygy ideal of %. The mapping zi ." #&i induces an isomorphismof rings

C[Y ] = C[z0, . . . , zq]/I %" C[#&0 , . . . ,#&q ].

In particular Y is irreducible. Identifying C[Y ] with C[#&0 , . . . ,#&q ],

the dominant rational mapping #! : P! ! Y corresponds to the fieldextension

C(##1 , . . . ,##q ) $ C(x1, . . . , x!).

The a!ne ring C[Y ] is isomorphic to a localization of the ring of Laurentpolynomials C[#±1

&0, . . . ,#±1

&q].

If " ! D % {0}, we write " = "a =)q

i=0 ai"&i =)n

i=1 ai"#i witha = (a0, . . . aq) ! Cq+1 % {0}, satisfying

)qi=0 ai = 0. Note that any

p-fold wedge product % = "#i1' · · · ' "#ip is a cocycle for ".

The one-form)q

i=0 aid log(zi) ! &(Cq+1) is C"-invariant, and con-tracts trivially along the Euler vector field, so it descends to a well-defined rational one-form on Pq. This form restricts to a one-form in&(Y ) which we denote by -a. Since Y $ (C")q, -a is regular on Y .Note that

)qi=0 aid log(zi) = d logµa, where µa =

!qi=0 z

aii is a master

function for the arrangement of coordinate hyperplanes in Pq.We show that the zeros of -a pull back to zeros of "a.

Lemma 3.4. Let x ! U and y = #!(x) ! Y . Then "a(x) = 0 ifand only if -a(y) ! ker(#"

!)y.

Proof. We have

"a =q%

i=0

aid log#&i = d logq$

i=0

#ai&i,= #"

!(d logq$

i=0

zaii ) = #"

!(-a).

The result follows upon localization at y. Q.E.D.

Note that

#"!(-a) =

q%

i=0

!%

j=0

ai#&i

'#&i

'xjdxj .

Then -a(y) ! ker(#"!)y if and only if

6a0y0

· · · aq

yq

7lies in the left null

space of the Jacobian of #!.Let Sing(#!) denote the singular locus of #!, and Sing(Y ) the sin-

gular locus of Y . Let S! = Sing(#!) + #!1! (Sing(Y )) $ U .

Theorem 3.5. V ("a) contains #!1! (V (-a)), and V ("a)%#!1(V (-a))

is a subset of S!.

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Proof. The first statement is immediate from Lemma 3.4. If #!

is nonsingular at x ! U then the Jacobian of #! attains its maximalrank p = dim(Y ) at x. If in addition Y is nonsingular at y = #!(x),then dim(&(Y )y) = p. Then ker(#"

!)y = 0. The second statement thenfollows from Lemma 3.4. Q.E.D.

Corollary 3.6. Suppose D is a rational singular subspace of A1,with integral basis %. Let p be the rank of D. If d log(#) ! D thencrit(#) $ S! or codim(crit(#)) # p.

Proof. Write " = d log(#) =)q

i=1 ai"#i . The hypothesis impliesdimY = p, so V (-a) is empty or has codimension at most p in Y .In the first case V (") $ S! by the preceding theorem. Otherwise,V (") / #!1

! (V (-a)) has codimension at most p. Q.E.D.

Corollary 3.7. Suppose D is a rational singular subspace of A1,with integral basis %. If d log(#) ! D, then crit(#) % S! is a union offibers of #!.

The fibers of #! are intersections of level sets of the rational masterfunctions ##i , for 1 # i # q.

We have not used the assumption that {+1, . . . , +q} is linearly inde-pendent, i.e., that the dimension of D is strictly greater than p. Thishypothesis rules out a trivial case.

Proposition 3.8. Suppose a (= 0. Then -a is not identically zeroon Y .

Proof. If -a is zero on Y , then

#"!(-a) =

q%

i=0

aid log(#&i) =q%

i=0

ai"&i =q%

i=1

ai"#i

is zero on U . This contradicts the assumption that {"#1 , . . . ,"#q} is abasis for D. Q.E.D.

A singular subspace of rank 1 is called an isotropic subspace of A1.

Corollary 3.9. Suppose % is an integral basis of an isotropic sub-space of A1. Then

(i) if " = d log(#) ! D then Sing(#!) $ crit(#).(ii) if 0 (= " = d log(#) ! D, then the components of crit(#)% S!

are disjoint hypersurfaces in U .

Proof. Since dim(Y ) = 1, the Jacobian of #! vanishes identicallyat points of Sing(#!). Then Sing(#!) $ V (") = crit(#) by Lemma 3.4.

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If a (= 0, then -a doesn’t vanish identically on Y by Proposition 3.8.Then V (-a) is zero-dimensional. Assertion (ii) follows from Theorem 3.5.

Q.E.D.

Corollary 3.9(i) can be used to locate singular fibers in Ceva pencils[10, Def. 4.5]—see Example 3.17.

3.2. Zeros of -aWe apply the method of Lagrange multipliers to find the zeros of -a.

The argument applies even if Y is singular. Fix a set of homogeneousgenerators {P1, . . . , Pr} of the defining ideal I = I! $ S of Y . Write 'jfor $

$zj. Let

J! =4'jPi

5

be the Jacobian of (P1, . . . , Pr). The rank of J! at a nonsingular pointy ! Y is equal to q % p, the codimension of Y .

Lemma 3.10. Let y ! Y . Then y ! V (-a) if and only if6a0y0

· · · aq

yq

7

is an element of the row space of J!(y).

Proof. The one-form d logµa =)q

i=0 aid log(zi) ! &(Pq) restrictsto -a ! &(Y ). There is an exact sequence of C[Y ]-modules

I/I2d%" C[Y ]0C[Pq ] &[Pq] " &[Y ] " 0,

where d is given by right multiplication by J! [9, Sec. 16.1]. Localiza-tion at the maximal ideal corresponding to y preserves exactness of thissequence, so -a(y) ! &(Y )y vanishes if and only if -a(y) is in the imageof d. Q.E.D.

For each i 1 0, let Fitti(a, I) be the variety in Pq defined by the(q + 1% i)) (q + 1% i) minors of the (r + 1)) (q + 1) matrix

(3.1)

,

222-

a0/z0 · · · aq/zq'0P1 · · · 'qP1...

. . ....

'0Pr · · · 'qPr

.

333/.

The ideal Fitti(a, I) is independent of the choice of generating set forI. Similarly, let Fitti(J!) denote the variety in Pq defined by the (q +1% i)) (q + 1% i) minors of J!. Let Y reg = Y % Sing Y . Then Y reg =Y * Fittp(J!)% Fittp+1(J!), where p = dimY .

If Y is smooth, then V (-a) is a Fitting variety. More generally:

Corollary 3.11. V (-a) * Y reg = Fittp(a, I) * Y reg.

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Proof. At any point of Y reg the rank of J! is equal to q % p =

codimY , so the rank of matrix (3.1) is at least q%p. Then6a0y0

· · · aq

yq

7

is in the row space of J(y) if and only if (3.1) has rank q% p, if and onlyif all (q % p+ 1)) (q % p+ 1) minors vanish. Q.E.D.

Remark 3.12. In fact, the intersections Y * Fitti(J!), p # i # qdetermine a stratification of Y by locally-closed subvarieties, and V (-a)coincides with Fitti(a, I) on the stratum Fitti(J!)%Fitti+1(J!), by thesame argument.

In view of Proposition 2.1 we study the zeros of cocycles for -a.Since dimY = p, every % ! &p(Y ) is a cocycle for -a. For 0 # i # q, set-i =

dyi

yi, so that -a =

)qi=0 ai-i =

)qi=1 ai(-i%-0) =

)qi=1 d log(yi/y0).

Proposition 3.13. The intersection*

{V (+) | + ! &pC(Y ), -a ' + = 0}

is contained in Sing(Y ).

Proof. For I = {i1 < · · · < ip}< $ {1, . . . , q}, consider the p-form

+I = (-i1 % -0) ' · · · ' (-ip % -0).

Then -a ' +I = 0. But dimY = p, so at each point of Y reg, +I must benonzero for some I. Q.E.D.

3.3. The case q = p+ 1

Suppose D is a singular subspace of rank dim(D)% 1, with integralbasis % = {"#1 , . . . ,"#q}. Then Y is defined by a single homogeneouspolynomial P (z0, . . . , zq). This hypothesis holds for all the examples weknow, with one exception: the Hessian arrangement, which supports arational singular subspace of dimension three and rank one—see Exam-ples 2.8 and 4.11.

Consider the rational mapping

(3.2) . = [z0'0P : · · · : zq'qP ] : Pq ! Pq.

This map has poles along Sing(Y ) and Sing(Y * CI) where CI is thecoordinate subspace zi = 0, i ! I. It is regular on Y reg * (C")q. ByEuler’s formula,

)qj=0 zj'jP = deg(P )P , so the image of Y under . is

contained in the hyperplane)q

j=0 zj = 0.

Proposition 3.14. Suppose Y is the hypersurface given by P = 0,and . is as given in (3.2). Then

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(i) V (-a) (= , if and only if a ! .(Y ).(ii) If a ! .(Y ) then V (-a) = .!1(a).(iii) For generic a ! .(Y ), V (-a) is nonempty and every compo-

nent has codimension equal to dim .(Y ).

Proof. The first two assertions follow from Lemma 3.10, and thethird follows from the second. Q.E.D.

Corollary 3.15. Suppose

(i) #! is nonsingular on U ,(ii) Y is nonsingular and intersects the coordinate hyperplanes

transversely,(iii) dim .(Y ) = dimY , and(iv) #!(U) = Y * (C")q.

If d log(#) =)q

i=1 ai"#i and ai (= 0 for all i, then crit(#) is nonemptyand every component has codimension q % 1.

Proof. We have observed that .(Y ) is contained in the hyperplane' defined by

)qi=0 zi = 0. The second hypothesis ensures that Sing(Y *

CI) is empty for every I $ {0, . . . , q}. Then . is regular on Y . By thethird condition, dim .(Y ) = q % 1 = dim'. Since ' is irreducible, weconclude .(Y ) = '. Since Y = Y * (C")q by (iii), ' * (C")q = .(Y ).The result then follows from Proposition 3.14 and Theorem 3.5, since(i) and (ii) imply S! = , and the fibers of #! all have codimensionq % 1. Q.E.D.

The hypotheses in Corollary 3.15 are satisfied in many examples.The third condition is automatic in case q = 2. The last two conditionshold in every example we know where (i) and (ii) hold.

3.4. Examples

Example 3.16 (Example 2.6, continued). Let D be the rationalsingular subspace of A1 &= C5 with basis % = {"010%"100, "001%"100}.

We have

#! = [#100 : #010 : #001] = [x(y % z) : y(x% z) : z(x% y)].

#! is nonsingular on U , and Y = Y * (C")5. The components of #!

satisfy the homogeneous relation #100%#010+#001 = 0, so Y is the linez0 % z1 + z2 = 0 in P2. Corollary 3.15 implies crit(#abc) is nonempty ifand only if a+ b+ c = 0 and a, b and c are nonzero. In this case,

crit(#abc) = (. 2 #!)!1([a : b : c]) = #!1

! ([a : %b : c]),

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that is, crit(#abc) is given by

[x(y % z) : y(x% z) : z(x% y)] = [a : %b : c], or, equivalently

[x(y % z) : y(z % x)] = [a : b]

as we found earlier. It has codimension one in P2. In this example themap #! has connected generic fiber, hence crit(#abc) is connected. Ifa, b, or c is zero, and a+ b+ c = 0, then crit(#abc) is empty.

The basis % above has special properties that resulted in the linearsyzygy of master functions: we will revisit this in the next section. Byway of comparison, consider the basis %% = {"120 % "012,"300 % "012}.Then

#!! = [#010#2001 : #100#

2010 : #3

100]

= [yz2(x% y)2(x% z) : xy2(x% z)2(y % z) : x3(y % z)3].

A Macaulay 2 calculation [12] shows that #!! is nonsingular on U . Usingthe identity #100 % #010 + #001 = 0, one finds that the Zariski closureY % of Y % = #!!(U) is defined by

z31 % z20z2 % 4z0z1z2 % 2z21z2 + z1z22 = 0.

Then Y % is an irreducible cubic with a node at [z0 : z1 : z2] = [%2 :1 : %1]. This is a point of Y %. It is also in 8#!!(E), where 8#!! is thelift of #!! to the blow-up of P2 at the four base points, and E is theexceptional divisor over [0 : 1 : 0].

The image of Y % under . misses the three points [0 : %1 : 1], [%2 :1 : 1], and [%2 : 3 : %1], corresponding to the three one-forms "100 %"010, "100 % "001, and "010 % "001 in D that have empty zero locus. Inparticular Y % (= Y % * (C")5.

Example 3.17. Let A be the arrangement with defining equationQ = Q0Q1Q2 where

Q0 = (x+ z)(2x% y % z)(2x+ y % z)

Q1 = (x% z)(2x+ y + z)(2x% y + z), and

Q2 = (y + z)(y % z)z

The image of #! = [Q0 : Q1 : Q2] : P2 ! P2 is the line z0%z1+2z2 =0. The fibers are the cubics passing through nine points, three on each ofthree concurrent lines—A is a specialization of the Pappus arrangement.One of these cubics is x(4x2% y2% 3z2) = 0. Although Y is smooth, #!

is singular at two points of U , given by x = y2 + 3z2 = 0. These twopoints lie in crit(Qa

0Qb1Q

c2) for every a, b, c with a+ b+ c = 0.

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Similarly, if A the subarrangement of the Hessian arrangement (2.8)defined by P1P2P3 = 0, then every critical set crit(P a

1 Pb2P

c3 ), a+b+c = 0,

contains the three points [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1] of U wherethe fourth special fiber xyz = 0 is singular. The map #! = [P1 :P2 : P3] is singular at these points, although Y is smooth, given by*P1 + *2P2 + P3 = 0. (See also Example 4.11.)

Remark 3.18. In fact, Corollary 3.9 can be used to detect Cevapencils [10] (see 4.4). For instance the master function

# =x(y % z)

y(x% z)

has critical points [0 : 1 : 0] and [1 : 1 : 0], the singular points of thethird completely decomposable fiber in Example 2.6.

Example 3.19 (Example 2.7, continued). In this example, the one-form " has no decomposable 2-cocycle. Indeed there are no singularsubspaces of A1 of rank p = 1 or p = 2. For p = 1 this holds becauseA is 2-generic. For p = 2 one can verify the statement computationallyusing the approach of [15]; in [2] we give a combinatorial argument basedon Theorem 5.4. Setting

( = [x

y + z + w:

y

x+ z + w:

z

x+ y + w:

w

x+ y + z] : P3 ! P3,

we have " = d log(#) = ("(-) where

- = a d log(y0) + bd log(y1) + c d log(y2) + d d log(y3).

The map ( is dominant. The one-dimensional critical locus of # is afiber of a di"erent map

[x(y + z + w) : y(x+ z + w) : z(x+ y + w) : w(x+ y + z)].

§4. Linear dependence among master functions

In this section we consider a singular subspace D with an integralbasis % such that Y! is linear, i.e., the syzygy ideal I! is generated byhomogeneous linear forms. We saw this phenomenon in Example 3.16.We start with a trivial example that will be useful for what follows.

4.1. Example: equations for the critical locus

Suppose A is an essential arrangement of n + 1 hyperplanes in P!,with n > #. Then D = A1 is a singular subspace, with integral basis

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{"i % "0 | 1 # i # n}. The corresponding rational mapping is

# = #! = [&0 : · · · : &n] : P! ! Pn,

and Y = Y ! is a linear subvariety of Pn. The reader will recognizethat this is the usual identification of a labelled vector configuration,(&0, . . . ,&n), with a point in the Grassmannian of #-planes in Pn. Letus denote Y ! by LA. (This is an abuse of notation; Y ! depends on thechoice of defining forms &i.)

Most of the results of the previous section are vacuous in this situ-ation, but Lemma 3.10 tells us something:

Theorem 4.1. Let B =4bij

5be an (# + 1) ) (n + 1) matrix such

that LA is the kernel of B. Then, for any ! ! Cn, the critical locus of#" is defined by the

'n+1!+1

(equations

%

i#I

)(i, I)bI!{i}!i

&i(x)= 0

where I ranges over the subsets of {0, . . . , n} of size #+1, the coe!cientbJ is given by bJ = det

4bij | j ! J

5, and )(i, I) = ±1, depending on the

position of i in I.

Proof. The linear forms defined by the rows of B generate thesyzygy ideal I!. Then the Jacobian J! is equal to B. With the observa-tion that S! = ,, setting a = ! in Lemma 3.10 and applying Theorem 3.5yields the claim. Q.E.D.

The columns of the matrix B above define a realization of the ma-troid dual to the matroid of A. In Example 2.6,

B =

,

-0 %1 1 0 0 1%1 0 1 0 1 0%1 1 0 1 0 0

.

/ .

(The matroid of A is self-dual.) Theorem 4.1 says that crit(#abc) isdefined by the 4) 4 minors of

,

22-

ax

by

cz

cx!y

bx!z

ay!z

0 %1 1 0 0 1%1 0 1 0 1 0%1 1 0 1 0 0

.

33/ .

These 15 equations reduce to the single equation found in Example 2.6.

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4.2. Linear hypersurfaces

Suppose Y is a linear and p = rank(D) = q % 1, i.e., Y is a linearhyperplane in Pq. Let P (z) =

)qj=0 bjzj be a generator for the syzygy

ideal. It is no loss to assume bj (= 0 for all j, or equivalently, Y is notcontained in any coordinate hyperplane. Otherwise some proper subsetof {#&0 , . . . ,#&q} is linearly dependent.

Proposition 4.2. Suppose Y is a hyperplane not contained in anycoordinate hyperplane in Pq. Let ! =

)qi=0 ai,i. Then for generic a

satisfying)q

i=0 ai = 0, crit(#")%S! is nonempty and every componentof crit(#")% S! has codimension equal to the rank of D.

Proof. The syzygy ideal I! is generated by a linear polynomialP (z) =

)qj=0 bjzj , and bj (= 0 for 0 # j # q by hypothesis. The map

. = [b0z0 : · · · : bqzq] of (3.2) is an automorphism of Pq since all of the bjare nonzero. Consequently, . maps the hyperplane Y isomorphically tothe hyperplane ' defined by

)qi=0 zi = 0. The result then follows from

Proposition 3.14. Q.E.D.

4.3. The general case

Suppose the singular subspace D $ A1 has an integral basis % forwhich Y = Y ! is a linear variety in Pq. Choose a linear isomorphism( : Pp " Y , given by a (q + 1) ) (p + 1) matrix B =

4bij

5. Assume D

is not contained in any coordinate hyperplane. Then the intersectionsof Y with the coordinate hyperplanes in Pq determine an essential ar-rangement B of q + 1 not necessarily distinct hyperplanes in Pp, withdefining forms /i(x) =

)pi=0 bijxj , for 0 # i # q. By construction, the

subspace LB of Pq, as described in Section 4.1, is equal to Y .

Theorem 4.3. Suppose Y ! = LB is a linear subspace not con-tained in any coordinate hyperplane in Pq, and Y = Y * (C")q. Let! =

)qi=0 ai,i with

)qi=0 ai = 0. Let (a =

!qi=0 /

aii be the master

function on the complement of the arrangement B in Pp correspondingto a. Then

(i) crit(#")% S! (= , if and only if crit((a) (= ,,(ii) crit(#")% S! = #!1

! (.(crit((a))), and(iii) codim(crit(#")% S!) # codim crit((a).

Proof. Just as in Section 4.1, the one-form -a on Y pulls backto d log((a) under the isomorphism (, and the assertions follow fromProposition 3.14. Q.E.D.

Example 4.4 (Example 2.6, continued). We saw in Example 3.16that the variety Y is given by z0%z1+z2 = 0 in P2, and Y = Y * (C")2.

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We can take ( : P1&=%%" Y to be given by the matrix

B =

,

-1 01 10 1

.

/ .

Then associated arrangement B consists of three points [1 : 0], [1 : 1],[0 : 1] in P1. The complement of B has Euler characteristic %1, so ageneric B-master function (abc has a single nondegenerate critical point.A computation shows that this holds if a, b, and c are nonzero. We reachthe same conclusion as before, that crit(#abc) has codimension one.

4.4. Multinets and codimension-one critical sets

Next we use the main result of [10] to give a complete descriptionof crit(#) for any " = d log(#) ! R1(A). As we observed earlier, if" ! R1(A), then " has a nontrivial decomposable 1-cocycle %. Thestatement that "'% = 0 means, for each x ! U, {"(x),%(x)} is linearlydependent. Then there are functions a and b on U such that a(x)"(x)+b(x)%(x) = 0 for all x ! U . This implies V (") contains V (b) % V (a),which is a hypersurface unless it is empty. It was this observation thatled to the current research.

According to [14], the maximal isotropic subspaces D of A1 of di-mension at least two are the components of R1(A), and they intersecttrivially. By [10, Theorem 3.11], such a component has an integral basis% = ("#1 , . . . ,"#q ), with the property that the corresponding polynomialmaster functions #&0 , . . . ,#&q are all collinear in the space of degree d

polynomials. Then Y = Y ! is a line in Pq. The homogenized basis{"&0 , . . . ,"&q} corresponds to the characteristic vectors ,i of the blocksin a multinet structure on a subarrangement of A, as defined below.

For X $ P!, write AX = {H ! A | X $ H}. A rank-two flat of Ais a subspace X of the form H * K for some H,K ! A, H (= K. If Pis a partition of A, the base locus of P is the set of rank-two flats of Aobtained by intersecting hyperplanes from di"erent blocks of P.

Definition 4.5. A (q + 1, d)-multinet on A is a pair (P,m) whereP is a partition {A0, . . . ,Aq} of A into q+1 blocks, with base locus X ,and m : A " Z>0 is a multiplicity function, satisfying

(i))

H#Aim(H) = d for every i.

(ii) For each X ! X ,)

H#Ai'AXm(H) = nX for some integer

nX , independent of i.(iii) For each i,

#Ai %

#X is connected.

The third condition says that P cannot be refined to a (q%, d)-multinet with the same multiplicity function, with q% > q + 1. Given a

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multinet on A, let ,i =)

Hi#Aim(Hi)ei, for 0 # i # q, and +i = ,i% ,0

for 1 # i # q. We call ,0, . . . , ,q the characteristic vectors of (P,m). Wehave the following results from [10].

Theorem 4.6 ([10, Corollary 3.12]). Suppose D is a maximal iso-tropic subspace of A1 of dimension q 1 2. Then there is a subarrange-ment A% of A and a (q+1, d)-multinet on A% whose characteristic vectors,0, . . . , ,q yield an integral basis "#1 , . . . ,"#q of D.

Theorem 4.7 ([10, Theorem 3.11]). Suppose ,0, . . . , ,q are thecharacteristic vectors of a (q+1, d)-multinet structure on A. Then eachof the master functions #&i , i 1 2, is a linear combination of #&0 and#&1 . Moreover every fiber of the mapping [#&0 : #&1 ] : P! ! P1 is con-nected.

Given this result, the analysis of critical sets proceeds exactly as inthe Example 4.4. First, we need a lemma about isolated critical points.

Lemma 4.8. Suppose A is an a!ne arrangement of n hyperplanesin C!, and W is a nonempty Zariski-open subset of C!. Then there is anonempty Zariski-open subset L of Cn such that W * crit(#") consistsof |$(U)| points, for each ! ! L.

Proof. By [20, Theorem 1.1], there is a nonempty Zariski-open sub-set L% of Cn for which crit(#") is isolated and consists of |$(U)| points,for ! ! L%. Let

$ = {(!, v) ! Cn ) U : ""(v) = 0} ,

an n-dimensional smooth complex variety by [20, Prop. 4.1]. Let 0ifor i = 1, 2 denote its projections onto Cn and U , respectively. Then0!12 (U *W ) and 0!1

1 (L%) are each nonempty Zariski-open subsets of $,as is their intersection Z. Then 01(Z) is a finite union of locally-closedsubsets of Cn—see [13, Exercise 3.19]. Since Z is dense in $, 01(Z) isdense in Cn. Hence 01(Z) contains a Zariski-open subset L, which hasthe required property. Q.E.D.

Theorem 4.9. Suppose " = d log(#) ! R1(A), and D is the max-imal isotropic subspace of A1 containing ". Let % be the integral basisof D arising from the associated multinet. Then

(i) For every " ! D, Sing(#!) $ crit(#), and,(ii) For generic " ! D, crit(#)%Sing(#!) is a union of dim(D)%1

connected smooth hypersurfaces of the same degree.

Proof. Write q = dim(D) and let ,0, . . . , ,q be the characteris-tic vectors of the (q + 1, d)-multinet corresponding to D. Write " =

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)qi=0 ai"&i . By the preceding theorem, for each 2 # k # q, there is a

linear relation #&k = bk#&0 + ck#&1 . Then Y is a line in Pq. The firstassertion follows from Corollary 3.9.

We can choose the isomorphism ( : P1 &=%%" Y 3 Pq to be given bythe matrix

B =

,

22222-

1 00 1b2 c2...

...bq cq

.

33333/.

The corresponding arrangement B consists of q + 1 distinct points inP1. The Euler characteristic of the complement of B is 1% q. Then forgeneric a the B-master function (a has q % 1 isolated, nondegeneratecritical points in Y * (C")q. In fact, since Y is dense in Y * (C")q, weapply Lemma 4.8 to see that, for generic a, (a has q % 1 critical pointsin Y . Then Corollary 3.7 implies crit(#) is the union of q % 1 fibers of#!.

The projection Pq ! P1 along z0 = z1 = 0 restricts to an isomor-phism on Y . Then the last statement of Theorem 4.7 implies the fibersof #! are connected. These fibers are given by [#&0 : #&1 ] = [a0 : a1].Since the #&i have degree d, crit(#!) is a union of q % 1 connected hy-persurfaces of degree d. The generic fiber of #! is smooth by Bertini’sTheorem [13, Corollary III.10.9]. Q.E.D.

Corollary 4.10. For generic " = d log(#) ! R1(A), the number ofconnected components of crit(#") is equal to the dimension of H1(A·,").

Example 3.17 shows that crit(#) need not be smooth or irreduciblefor all " = d log(#) ! R1(A).

Example 4.11. By [22, 31], the maximum number of blocks in amultinet is equal to four. The only known example with four blocks is themultinet on the Hessian arrangement corresponding to the Hesse pencil,Example 2.8. The factors of the polynomial master functions P0 =xyz, P1, P2, P3 define the blocks of a multinet on A with all multiplicitiesequal to one. These master functions satisfy two linear syzygies:

P2 = 3(1% *)P0 + P1

P3 = 3(1% *2)P0 + P1.

Then the variety Y ! corresponding to the basis

% = {"100,"010,"001}

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Fig. 2. A rank-four matroid with a linear syzygy of master functions

is the line given by z2 = 3(1 % *)z0 + z1, z3 = 3(1 % *2)z0 + z1 in P3,which meets the coordinate hyperplanes in the four points correspondingto the singular fibers. The corresponding arrangement B consists of fourpoints in general position in P1, given by the rows of the matrix

,

22-

1 00 1

3(1% *) 13(1% *2) 1

.

33/ .

The complement of B has Euler characteristic %2, hence a genericB-master function has two isolated critical points. Then, for generica = (a1, a2, a3), the critical locus of the A-master function # = #a1a2a3

has two components and codimension one, as found by direct calculationin Example 2.8. This example shows that Theorem 4.8(ii) and Corollary4.9 may not hold under the weaker hypothesis that ai (= 0 for all i.

Here is a rank-four example, that has appeared in di"erent form inthe lecture of A. Libgober in this volume.

Example 4.12. LetA be the arrangement with defining polynomial

Q = (x+ y)(x% y)(y + z)(y % z)(z + w)(z % w)(w + x)(w % x),

with the hyperplanes numbered according to the order of factors in Q.Then A is a 2-generic subarrangement of the Coxeter arrangement oftype D4. Up to lattice-isotopy, the dual projective point configurationconsists of the eight vertices of a cube—see Figure 2. Let D be thesubspace of A1 with basis

% = {("0+"1)% ("6+"7), ("2+"3)% ("6+"7), ("4+"5)% ("6+"7)}.

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Then D is a rational singular subspace of rank two; if " ! D, thenH1(A·,") = 0 and H2(A·,") &= D/C" has dimension 2. Y ! is thelinear hyperplane in P3 defined by z0 + z1 + z2 + z3 = 0, reflecting thelinear syzygy of polynomial master functions

(x+ y)(x% y) + (y + z)(y % z) + (z + w)(z % w) + (w + x)(w % x) = 0.

From Proposition 4.2 we see that

# =

0x2 % y2

w2 % x2

1a10y2 % z2

w2 % x2

1a20z2 % w2

w2 % x2

1a3

has nonempty critical set of codimension two in P3, for generic (a1, a2, a3).

§5. The rank condition

We are left with the problem of finding rational singular subspacesof A1. The theory of multinets gives a method to find such subspaces ofrank one. In this section we give a combinatorial condition for a set %of linearly independent integral one-forms to span a singular subspaceof A1 of arbitrary rank, using tropical implicitization and nested sets.

5.1. Tropicalization

The tropicalization of a projective variety V in Pq is a polyhedral fantrop(V ) in tropical projective space TPq = Rq+1/R(1, . . . , 1), associatedto a homogeneous defining ideal I of V . If V is a hypersurface withdefining equation f = 0, then trop(V ) is the image in TPq of the unionof the cones of codimension at least one in the normal fan of the Newtonpolytope of f . In general, trop(V ) is the image of the union of the conesof codimension at least one in the Grobner fan of I. The set trop(V )arises geometrically from the lowest-degree terms in Puiseux expansionsof curves lying in V . See [7] and the references therein for backgroundon tropical varieties. See [21] for matroid terminology.

We will need several results from tropical geometry. The first is atheorem of Bieri and Groves [3].

Theorem 5.1. The maximal cones in trop(V ) have dimension equalto dim(V ).

If V is an #-dimensional linear subvariety of Pn, given as the columnspace of a matrix R, then the tropicalization trop(V ) depends only onthe dependence matroid G on the rows of R. In our setting the rows ofR give the defining forms of a hyperplane arrangement A. The matroid

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polytope '(G) of G is the convex hull of the set

9%

i#B

ei | B is a basis of G:.

The tropicalization trop(V ), called the Bergman fan of G, is the image inTPn of the union of the cones of codimension at least one in the normalfan of '(G). We denote it by B(G).

In [11] the Bergman fan is described in terms of nested set cones. LetG be the set of proper connected (i.e., irreducible) flats of G. These arethe flats corresponding to the dense edges of the projective arrangementA. A collection S = {X1, . . . , Xp} of subsets of G is a nested set if, forevery set T of pairwise incomparable elements of S, the join

;T is not

an element of G. The nested sets form a simplicial complex ' = '(G),the nested set complex, which is pure of dimension r = # % 1. It is thecoarsest of a family of nested set complexes, obtained by replacing Gwith larger “building sets.” All of these complexes are subdivided bythe order complex of the poset of nonempty flats of G.

If S $ A, set eS =)

Hi#S ei. The nested set fan N(G) is the imagein TPn of the union of the cones generated by

{eS | S ! S}

for S ! '(G). From [11] we have the following result.

Theorem 5.2. The nested set fan N(G) subdivides the Bergmanfan B(G).

5.2. Singular subspaces

Let A be an arrangement in P! with homogeneous defining linearforms &0, . . . ,&n. Let G be the underlying matroid of A, the dependencematroid on {&0, . . . ,&n}. Suppose D is a rational subspace of A1(A),with integral basis % = {"#1 , . . . ,"#q}. We identify % with the q)(n+1)matrix of integers

4+ij

5, and recall that

)nj=0 +ij = 0 for 1 # i # q. Let

Y ! be the Zariski closure of the image of the associated rational map#! = [1 : ##1 : · · · : ##q ] : P! ! Pq.

The main observation is that #! can be factored as a linear mapfollowed by a monomial map. Assume A is essential, and let

& =4&0 : · · · : &n

5: P! " Pn.

Let µ = µ! : Pn ! Pq be given by

µ([t0 : · · · : tn]) = [1 : t#1 : · · · : t#q ],

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where we use the usual vector notation for monomials: t(i0,...,in) =ti00 · · · tinn . Then the following diagram commutes.

Pn!!

µ!

!!!!!

!!!!

!

P! "" "! ""

'

##"""""""""" Pq

In this situation the diagram tropicalizes faithfully, in the followingsense.

Theorem 5.3 ([7, Theorem 3.1]). The tropicalization trop(Y !) isequal to the image of the Bergman fan B(G) under the linear map

TPn " TPq

with matrix %.

We obtain the following characterization. Write % =4%0| · · · |%n

5

with %j ! Zq for each j. For S = {Hj1 , . . . , Hjk} $ A, let %S =)kr=1 %jr .

Theorem 5.4. The subspace D is singular if and only if the rankof the matrix

%S =4%S1 | · · · |%S""1

5

is less than q, for each maximal nested set S ! N(G). In this case therank of D is the maximal rank of %S for S ! N(G).

Proof. The subspace D is singular if and only if dimY ! < q. ByTheorem 5.1, this occurs if and only if dim trop(Y !) < q. The conesof trop(Y !) are images of the cones of B(G) under %, by Theorem 5.3.The linear hulls of the cones in B(G) are the images in TPn of the linearspans of the sets {eS | S ! S}, for S ! N(G), by Theorem 5.2. Since%(eS) = %S , the result follows. The last statement holds because therank of D is equal to dimY !. Q.E.D.

Example 5.5. Consider the arrangement of rank four with definingpolynomial

Q = xyz(x+ y + z)w(x+ y + w),

with hyperplanes ordered according to the given factorization of Q. Thedual point configuration consists of the six vertices of a triangular prismin P3.

For generic (a, b, c), the master function # = xay!azb(x + y +z)!bwc(x+y+w)!c has critical set of codimension two, and H2(A,") &=

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C for " = d log(#). Based on our other examples, one might suspectthat the subspace D with basis % = {"0 % "1, "2 % "3, "4 % "5} issingular. Among the nested sets of A is the set S = {0, 02, 024}, and

%S =

,

-1 1 10 1 10 0 1

.

/

does not have rank two. Then by Theorem 5.4, D is not singular. Infact, % = a · '("0"1"5) + b · '("2"3"5) is the unique 2-cocycle for ".% is trivial if a or b is zero, and our argument shows that % is notdecomposable if a and b are both nonzero.

In the forthcoming paper [2], Theorem 5.4 is used to derive combi-natorial conditions for p-generic arrangements to support singular sub-spaces of rank p. Using that approach one can show by combinatorialmeans that there are no singular subspaces of A1 of rank two in Exam-ple 5.5.

Theorem 5.4 also has the following corollary.

Corollary 5.6. If G1 and G2 are loop-free matroids on the groundset {1, . . . , n} and B(G1) = B(G2), then G1 = G2.

Proof. Let G be a loop-free matroid on {1, . . . , n}, with Orlik–Solomon algebra A· = A·(G). Let e1, . . . , en ! A1 denote the canonicalgenerators. Then S = {i1, . . . , ik} $ {1, . . . , n} is dependent in G ifand only if ei1 ' · · · ' eik = 0 in Ak. (This statement holds even if Ghas multiple points.) Equivalently, S is dependent if and only if thecoordinate subspace D $ A1 spanned by {ei1 , . . . , eik} is singular. ByTheorem 5.3, D is singular if and only if the image of the Bergman fanB(G) $ TPn under the projection TPn " TPS &= TPk!1 has dimensionless than k % 1. Thus B(G) determines G. Q.E.D.

Acknowledgements. This work was begun during the semester-longprogram on Hyperplane Arrangements at MSRI in Fall, 2004. The au-thors are grateful to the institute for support. We thank Bernd Sturmfelsfor pointing out the relevance of Theorem 5.3. We also thank an anony-mous referee for pointing out the phenomena illustrated in Example 3.17,leading us to correct an earlier version of Theorem 3.5.

References[ 1 ] V. Arnol’d, The cohomology ring of the group of dyed braids, Mat. Zametki,

5 (1969), 227–231, MR0242196.[ 2 ] C. Bibby, M. Falk and I. Williams, Decomposable cocycles for p-generic

arrangements, in preparation.

62104falk.tex : 2012/2/15 (10:4) page: 106


[ 3 ] R. Bieri and J. R. J. Groves, The geometry of the set of characters inducedby valuations, J. Reine Angew. Math., 347 (1984), 168–195, MR733052.

[ 4 ] E. Brieskorn, Sur les groupes de tresses, In: Seminaire Bourbaki,1971/72, 317 Lecture Notes in Math., Springer-Verlag, 1973, pp. 21–44,MR0422674.

[ 5 ] D. Cohen, G. Denham, M. Falk and A. Varchenko, Critical points andresonance of hyperplane arrangements, Canad. J. Math., 63 (2011), 1038–1057.

[ 6 ] D. Cohen and A. Varchenko, Resonant local systems and representations ofsl2, Geom. Dedicata, 101 (2003), 217–233, MR2017904.

[ 7 ] A. Dickenstein, E.-M. Feichtner and B. Sturmfels, Tropical discriminants,J. Amer. Math. Soc., 20 (2007), 1111–1133, MR2328718.

[ 8 ] A. Dimca, Singularities and Topology of Hypersurfaces, Universitext,Springer-Verlag, 1992, MR1194180.

[ 9 ] D. Eisenbud, Commutative Algebra with a View toward Algebraic Geome-try, Grad. Texts in Math., 150, Springer-Verlag, 1995, MR1322960.

[10] M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils ofplane curves, Compos. Math., 143 (2007), 1069–1088, MR2339840.

[11] E.-M. Feichtner and B. Sturmfels, Matroid polytopes, nested sets, andBergman fans, Port. Math., 62 (2005), 437–468, MR2191630.

[12] D. Grayson and M. Stillman, Macaulay 2: a software systemfor algebraic geometry and commutative algebra, available atwww.math.uiuc.edu/Macaulay2.

[13] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math., 52, Springer-Verlag, 1977, MR0463157.

[14] A. Libgober and S. Yuzvinsky, Cohomology of the Orlik–Solomon algebrasand local systems, Compos. Math., 121 (2000), 337–361, MR1761630.

[15] P. Lima-Filho and H. Schenck, E!cient computation of resonance vari-eties via Grassmannians, J. Pure Appl. Algebra, 213 (2009), 1606–1611,MR2517996.

[16] E. Mukhin and A. Varchenko, Critical points of master functions and flagvarieties, Commun. Contemp. Math., 6 (2004), 111–163, MR2048778.

[17] E. Mukhin and A. Varchenko, Norm of a Bethe vector and the Hessian of themaster function, Compos. Math., 141 (2005), 1012–1028, MR2148192.

[18] E. Mukhin and A. Varchenko, Miura opers and critical points of mas-ter functions, Cent. Eur. J. Math., 3 (2005), 155–182 (electronic),MR2129922.

[19] P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren Math.Wiss., 300, Springer-Verlag, 1992, MR1217488.

[20] P. Orlik and H. Terao, The number of critical points of a product of powersof linear functions, Invent. Math., 120 (1995), 1–14, MR1323980.

[21] J. Oxley, Matroid Theory, Oxford Univ. Press, New York, 1992,MR1207587.

[22] J. Pereira and S. Yuzvinsky, Completely reducible hypersurfaces in a pencil,Adv. Math., 219 (2008), 672–688, MR2435653.

62104falk.tex : 2012/2/15 (10:4) page: 107


[23] N. Reshetikhin and A. Varchenko, Quasiclassical asymptotics of solutions tothe KZ equations, In: Geometry, Topology, and Physics for Raoul Bott,Conf. Proc. Lecture Notes Geom. Topology, IV, International Press,Cambridge, MA, 1995, pp. 293–322, MR1358621.

[24] V. Schechtman and A. Varchenko, Arrangements of hyperplanes and Liealgebra homology, Invent. Math., 106 (1991), 139–194, MR1123378.

[25] I. Scherbak and A. Varchenko, Critical points of functions, sl2 representa-tions, and Fuchsian di"erential equations with only univalued solutions,Mosc. Math. J., 3 (2003), 621–645, MR2025276.

[26] R. Silvotti, On a conjecture of Varchenko, Invent. Math., 126 (1996), 235–248, MR1411130.

[27] A. Varchenko, Critical points of the product of powers of linear functionsand families of bases of singular vectors, Compos. Math., 97 (1995), 385–401, MR1353281.

[28] A. Varchenko, Special functions, KZ type equations, and representationtheory, CBMS Regional Conf. Ser. in Math., 98, Amer. Math. Soc. Prov-idence, RI, 2003, MR2016311.

[29] A. Varchenko, Quantum Integrable Model of an Arrangement of Hyper-planes, SIGMA, 7 (2011), 032, 55 pp.http://dx.doi.org/10.3842/SIGMA.2011.032.

[30] S. Yuzvinsky, Cohomology of the Brieskorn-Orlik-Solomon algebras, Comm.Algebra, 23 (1995), 5339–5354, MR1363606.

[31] S. Yuzvinsky, A new bound on the number of special fibers in a pencil ofcurves. Proc. Amer. Math. Soc., 137 (2008), 1641–1648, MR2470822.

Daniel C. CohenDepartment of Mathematics, Louisiana State UniversityBaton Rouge, LA 70803, USAE-mail address: [email protected]

Graham DenhamDepartment of Mathematics, University of Western OntarioLondon, ON N6A 5B7, CanadaE-mail address: [email protected]

Michael FalkDepartment of Mathematics and Statistics, Northern Arizona University,Flagsta", AZ 86011, USAE-mail address: [email protected]

Alexander VarchenkoDepartment of Mathematics, University of North Carolina at Chapel HillChapel Hill, NC 27599, USAE-mail address: [email protected]

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