ph.d. seminar, university of genoav - dima · 2017. 4. 6. · extremal behaviour in sectional...
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Extremal Behaviour in Sectional Matrices
Elisa Palezzato1
joint work with Anna Maria Bigatti1 and Michele Torielli2
1University of Genova, Italy2Hokkaido University, Japan
arXiv:1702.03292
Ph.D. Seminar,University of Genova
6 March 2017
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Extremal Behaviour in Sectional Matrices
1 PREQUEL
2 Introduction
3 Sectional matrix and its algebraic properties
4 Geometrical properties
5 Examples
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1 PREQUEL
2 Introduction
3 Sectional matrix and its algebraic properties
4 Geometrical properties
5 Examples
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From Wikipedia I
A Computer Algebra System
is a software program that allows computation over mathematicalexpressions in a way which is similar to the traditional manualcomputations of mathematicians and scientists.
The development of the computer algebra systems started in the secondhalf of the 20th century and this discipline is called �computer algebra�or �symbolic computation�.
Computer algebra systems may be divided in two classes:
The specialized ones are devoted to a speci�c part of mathematics,such as number theory, group theory, etc. [CoCoA, Macaulay2,Singular, ...]
General purpose computer algebra systems aim to be useful to auser working in any scienti�c �eld that requires manipulation ofmathematical expressions. [Matlab, Maple, Magma, ...]
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From Wikipedia II
A Gröbner basis
is a particular kind of generating set of an ideal in a polynomial ringover a �eld K[x1, . . . , xn].
Gröbner basis computation is one of the main practical tools for solvingsystems of polynomial equations and computing the images of algebraicvarieties under projections or rational maps.
Gröbner basis computation can be seen as a multivariate, non-lineargeneralization of both Euclidian algorithm for computing polynomialgreatest common divisors, and Gaussian elimination for linear systems.
What you need to compute a Gröbner Basis:
polynomial ring
monomial ordering
reduction algorithm
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1 PREQUEL
2 Introduction
3 Sectional matrix and its algebraic properties
4 Geometrical properties
5 Examples
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Hilbert Function
The computation of Hilbert function is available in most computeralgebra systems.
De�nition
Let K be a �eld of characteristic 0. Given a homogeneous ideal I inP = K[x1, . . . , xn], we de�ne the Hilbert function of I to be the function
HI(d) := dimK(Id).
Similarly, we de�ne the Hilbert function of P/I to be the function
HP/I(d) := dimK((P/I)d).
The Hilbert function is important in computational algebraic geometry,as it is the easiest known way for computing the dimension and thedegree of an algebraic variety de�ned by explicit polynomial equations.
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Hilbert Function
The computation of Hilbert function is available in most computeralgebra systems.
De�nition
Let K be a �eld of characteristic 0. Given a homogeneous ideal I inP = K[x1, . . . , xn], we de�ne the Hilbert function of I to be the function
HI(d) := dimK(Id).
Similarly, we de�ne the Hilbert function of P/I to be the function
HP/I(d) := dimK((P/I)d).
The Hilbert function is important in computational algebraic geometry,as it is the easiest known way for computing the dimension and thedegree of an algebraic variety de�ned by explicit polynomial equations.
Elisa Palezzato Ph.D. Seminar 6 March 2017 7 / 27
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Hilbert Function
The computation of Hilbert function is available in most computeralgebra systems.
De�nition
Let K be a �eld of characteristic 0. Given a homogeneous ideal I inP = K[x1, . . . , xn], we de�ne the Hilbert function of I to be the function
HI(d) := dimK(Id).
Similarly, we de�ne the Hilbert function of P/I to be the function
HP/I(d) := dimK((P/I)d).
The Hilbert function is important in computational algebraic geometry,as it is the easiest known way for computing the dimension and thedegree of an algebraic variety de�ned by explicit polynomial equations.
Elisa Palezzato Ph.D. Seminar 6 March 2017 7 / 27
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Hilbert Function
The computation of Hilbert function is available in most computeralgebra systems.
De�nition
Let K be a �eld of characteristic 0. Given a homogeneous ideal I inP = K[x1, . . . , xn], we de�ne the Hilbert function of I to be the function
HI(d) := dimK(Id).
Similarly, we de�ne the Hilbert function of P/I to be the function
HP/I(d) := dimK((P/I)d).
The Hilbert function is important in computational algebraic geometry,as it is the easiest known way for computing the dimension and thedegree of an algebraic variety de�ned by explicit polynomial equations.
Elisa Palezzato Ph.D. Seminar 6 March 2017 7 / 27
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Question
Can the Hilbert function characterize also some of the geometrical
behaviour of algebraic variety in the projective space?
Answer
In general no.
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Question
Can the Hilbert function characterize also some of the geometrical
behaviour of algebraic variety in the projective space?
Answer
In general no.
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Example in P2
x
y
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Example in P3
zx
y
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Question
Can we �nd an other algebraic invariant that characterize some of the
geometrical behaviour of algebraic variety in the projective space?
Answer
Yes, the sectional matrix.
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Question
Can we �nd an other algebraic invariant that characterize some of the
geometrical behaviour of algebraic variety in the projective space?
Answer
Yes, the sectional matrix.
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1 PREQUEL
2 Introduction
3 Sectional matrix and its algebraic properties
4 Geometrical properties
5 Examples
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Sectional Matrix
De�nition
Let K be a �eld of characteristic 0. Given a homogeneous ideal I inP = K[x1, . . . , xn], we de�ne the sectional matrix of I to be thefunction
MI(i, d) := dimK(I + (L1, . . . , Ln−i)/(L1, . . . , Ln−i))d
where L1, . . . , Ln−i are general linear forms, i = 1, . . . , n and d ≥ 0.
Similarly, we de�ne the sectional matrix of P/I to be the function
MP/I(i, d) := dimK(P/(I + (L1, . . . , Ln−i)/(L1, . . . , Ln−i)))d
=
(d+ i− 1
i− 1
)−MI(i, d)
where L1, . . . , Ln−i are general linear forms, i = 1, . . . , n and d ≥ 0.
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Sectional Matrix
De�nition
Let K be a �eld of characteristic 0. Given a homogeneous ideal I inP = K[x1, . . . , xn], we de�ne the sectional matrix of I to be thefunction
MI(i, d) := dimK(I + (L1, . . . , Ln−i)/(L1, . . . , Ln−i))d
where L1, . . . , Ln−i are general linear forms, i = 1, . . . , n and d ≥ 0.Similarly, we de�ne the sectional matrix of P/I to be the function
MP/I(i, d) := dimK(P/(I + (L1, . . . , Ln−i)/(L1, . . . , Ln−i)))d
=
(d+ i− 1
i− 1
)−MI(i, d)
where L1, . . . , Ln−i are general linear forms, i = 1, . . . , n and d ≥ 0.
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Strongly Stable Ideal
De�nition
Let I be a homogenous ideal in P = K[x1, . . . , xn]. We say that I is astrongly stable ideal if T = xi11 · · ·xinn ∈ I, then xi · T/xj ∈ I for alli < j ≤ max{k | ik 6= 0}.
Example
The ideal I = (x3, x2y, xy2, xyz) is not a strongly stable in Q[x, y, z]because x · xyz/y = x2z 6∈ I. The ideal I + (x2z) is strongly stable.
Remark
If I is a strongly stable ideal, then in the de�nition of the sectional
matrix we can take Li = xn−i+1.
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Strongly Stable Ideal
De�nition
Let I be a homogenous ideal in P = K[x1, . . . , xn]. We say that I is astrongly stable ideal if T = xi11 · · ·xinn ∈ I, then xi · T/xj ∈ I for alli < j ≤ max{k | ik 6= 0}.
Example
The ideal I = (x3, x2y, xy2, xyz) is not a strongly stable in Q[x, y, z]because x · xyz/y = x2z 6∈ I. The ideal I + (x2z) is strongly stable.
Remark
If I is a strongly stable ideal, then in the de�nition of the sectional
matrix we can take Li = xn−i+1.
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Strongly Stable Ideal
De�nition
Let I be a homogenous ideal in P = K[x1, . . . , xn]. We say that I is astrongly stable ideal if T = xi11 · · ·xinn ∈ I, then xi · T/xj ∈ I for alli < j ≤ max{k | ik 6= 0}.
Example
The ideal I = (x3, x2y, xy2, xyz) is not a strongly stable in Q[x, y, z]because x · xyz/y = x2z 6∈ I. The ideal I + (x2z) is strongly stable.
Remark
If I is a strongly stable ideal, then in the de�nition of the sectional
matrix we can take Li = xn−i+1.
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Generic Initial Ideal
Theorem (Galligo '74)
Let I be a homogeneous ideal in K[x1, . . . , xn], σ a term-ordering such
that x1 >σ x2 >σ · · · >σ xn. Then there exists a Zariski open set
U ⊆ GL(n) and a strongly stable ideal J such that for each g ∈ U ,LTσ(g(I)) = J .
De�nition
The strongly stable ideal J given in the previous Theorem will be calledthe generic initial ideal with respect to σ of I and it will be denoted byginσ(I). In particular, ginDegRevLex(I) is denoted by many authors withrgin(I).
Example
Consider the ideal I = (z5, xyz3) in Q[x, y, z], thenrgin(I) = (x5, x4y, x3y3).
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Generic Initial Ideal
Theorem (Galligo '74)
Let I be a homogeneous ideal in K[x1, . . . , xn], σ a term-ordering such
that x1 >σ x2 >σ · · · >σ xn. Then there exists a Zariski open set
U ⊆ GL(n) and a strongly stable ideal J such that for each g ∈ U ,LTσ(g(I)) = J .
De�nition
The strongly stable ideal J given in the previous Theorem will be calledthe generic initial ideal with respect to σ of I and it will be denoted byginσ(I). In particular, ginDegRevLex(I) is denoted by many authors withrgin(I).
Example
Consider the ideal I = (z5, xyz3) in Q[x, y, z], thenrgin(I) = (x5, x4y, x3y3).
Elisa Palezzato Ph.D. Seminar 6 March 2017 15 / 27
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Generic Initial Ideal
Theorem (Galligo '74)
Let I be a homogeneous ideal in K[x1, . . . , xn], σ a term-ordering such
that x1 >σ x2 >σ · · · >σ xn. Then there exists a Zariski open set
U ⊆ GL(n) and a strongly stable ideal J such that for each g ∈ U ,LTσ(g(I)) = J .
De�nition
The strongly stable ideal J given in the previous Theorem will be calledthe generic initial ideal with respect to σ of I and it will be denoted byginσ(I). In particular, ginDegRevLex(I) is denoted by many authors withrgin(I).
Example
Consider the ideal I = (z5, xyz3) in Q[x, y, z], thenrgin(I) = (x5, x4y, x3y3).
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Properties of the Sectional Matrix
Proposition
Let I be a homogeneous ideal in P = K[x1, . . . , xn] with minimal
generators of degree ≤ δ. Then
1 MI(n,−) coincides with the Hilbert function of I, andMP/I(n,−)with the one of P/I.
2 MI =Mrgin(I), andMP/I =MP/rgin(I).
3 MP/I(k, d+ 1) ≤∑k
i=1MP/I(i, d), for all k and d. If we have
MP/I(k, δ + 1) =∑k
i=1MP/I(i, δ), thenMP/I(s, d+ 1) =
∑si=1MP/I(i, d), for all s ≤ k and d ≥ δ.
4 If I is a strongly stable ideal (I = rgin(I)), thenMP/I(k, d+ 1) =
∑ki=1MP/I(i, d), for all d > δ and for all k.
5 If δ = reg(I), thenMP/I(k, d+ 1) =∑k
i=1MP/I(i, d), for alld > δ and for all k.
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Properties of the Sectional Matrix
Proposition
Let I be a homogeneous ideal in P = K[x1, . . . , xn] with minimal
generators of degree ≤ δ. Then1 MI(n,−) coincides with the Hilbert function of I, andMP/I(n,−)
with the one of P/I.
2 MI =Mrgin(I), andMP/I =MP/rgin(I).
3 MP/I(k, d+ 1) ≤∑k
i=1MP/I(i, d), for all k and d. If we have
MP/I(k, δ + 1) =∑k
i=1MP/I(i, δ), thenMP/I(s, d+ 1) =
∑si=1MP/I(i, d), for all s ≤ k and d ≥ δ.
4 If I is a strongly stable ideal (I = rgin(I)), thenMP/I(k, d+ 1) =
∑ki=1MP/I(i, d), for all d > δ and for all k.
5 If δ = reg(I), thenMP/I(k, d+ 1) =∑k
i=1MP/I(i, d), for alld > δ and for all k.
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Properties of the Sectional Matrix
Proposition
Let I be a homogeneous ideal in P = K[x1, . . . , xn] with minimal
generators of degree ≤ δ. Then1 MI(n,−) coincides with the Hilbert function of I, andMP/I(n,−)
with the one of P/I.
2 MI =Mrgin(I), andMP/I =MP/rgin(I).
3 MP/I(k, d+ 1) ≤∑k
i=1MP/I(i, d), for all k and d. If we have
MP/I(k, δ + 1) =∑k
i=1MP/I(i, δ), thenMP/I(s, d+ 1) =
∑si=1MP/I(i, d), for all s ≤ k and d ≥ δ.
4 If I is a strongly stable ideal (I = rgin(I)), thenMP/I(k, d+ 1) =
∑ki=1MP/I(i, d), for all d > δ and for all k.
5 If δ = reg(I), thenMP/I(k, d+ 1) =∑k
i=1MP/I(i, d), for alld > δ and for all k.
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Properties of the Sectional Matrix
Proposition
Let I be a homogeneous ideal in P = K[x1, . . . , xn] with minimal
generators of degree ≤ δ. Then1 MI(n,−) coincides with the Hilbert function of I, andMP/I(n,−)
with the one of P/I.
2 MI =Mrgin(I), andMP/I =MP/rgin(I).
3 MP/I(k, d+ 1) ≤∑k
i=1MP/I(i, d), for all k and d. If we have
MP/I(k, δ + 1) =∑k
i=1MP/I(i, δ), thenMP/I(s, d+ 1) =
∑si=1MP/I(i, d), for all s ≤ k and d ≥ δ.
4 If I is a strongly stable ideal (I = rgin(I)), thenMP/I(k, d+ 1) =
∑ki=1MP/I(i, d), for all d > δ and for all k.
5 If δ = reg(I), thenMP/I(k, d+ 1) =∑k
i=1MP/I(i, d), for alld > δ and for all k.
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Properties of the Sectional Matrix
Proposition
Let I be a homogeneous ideal in P = K[x1, . . . , xn] with minimal
generators of degree ≤ δ. Then1 MI(n,−) coincides with the Hilbert function of I, andMP/I(n,−)
with the one of P/I.
2 MI =Mrgin(I), andMP/I =MP/rgin(I).
3 MP/I(k, d+ 1) ≤∑k
i=1MP/I(i, d), for all k and d. If we have
MP/I(k, δ + 1) =∑k
i=1MP/I(i, δ), thenMP/I(s, d+ 1) =
∑si=1MP/I(i, d), for all s ≤ k and d ≥ δ.
4 If I is a strongly stable ideal (I = rgin(I)), thenMP/I(k, d+ 1) =
∑ki=1MP/I(i, d), for all d > δ and for all k.
5 If δ = reg(I), thenMP/I(k, d+ 1) =∑k
i=1MP/I(i, d), for alld > δ and for all k.
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Properties of the Sectional Matrix
Proposition
Let I be a homogeneous ideal in P = K[x1, . . . , xn] with minimal
generators of degree ≤ δ. Then1 MI(n,−) coincides with the Hilbert function of I, andMP/I(n,−)
with the one of P/I.
2 MI =Mrgin(I), andMP/I =MP/rgin(I).
3 MP/I(k, d+ 1) ≤∑k
i=1MP/I(i, d), for all k and d. If we have
MP/I(k, δ + 1) =∑k
i=1MP/I(i, δ), thenMP/I(s, d+ 1) =
∑si=1MP/I(i, d), for all s ≤ k and d ≥ δ.
4 If I is a strongly stable ideal (I = rgin(I)), thenMP/I(k, d+ 1) =
∑ki=1MP/I(i, d), for all d > δ and for all k.
5 If δ = reg(I), thenMP/I(k, d+ 1) =∑k
i=1MP/I(i, d), for alld > δ and for all k.
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Example
Let I be the zero-dimensional homogeneous ideal
(x2+y2−25z2, y4−3xy2z−4y3z+12xyz2−25y2z2+100yz3, xy3−16xyz2)
and rgin(I) =(x2, xy3, y4).
The sectional matrix of I is
0 1 2 3 4 5 . . .HI+〈L1,L2〉(d) =MI(1, d) : 0 0 1 1 1 1 . . .HI+〈L1〉(d) =MI(2, d) : 0 0 1 2 5 6 . . .
HI(d) =MI(3, d) : 0 0 1 3 8 14 . . .
The sectional matrix of P/I is
0 1 2 3 4 5 . . .HP/(I+〈L1,L2〉)(d) =MP/I(1, d) : 1 1 0 0 0 0 . . .
HP/(I+〈L1〉)(d) =MP/I(2, d) : 1 2 2 2 0 0 . . .
HP/I(d) =MP/I(3, d) : 1 3 5 7 7 7 . . .
Elisa Palezzato Ph.D. Seminar 6 March 2017 17 / 27
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Example
Let I be the zero-dimensional homogeneous ideal
(x2+y2−25z2, y4−3xy2z−4y3z+12xyz2−25y2z2+100yz3, xy3−16xyz2)
and rgin(I) =(x2, xy3, y4).
The sectional matrix of I is
0 1 2 3 4 5 . . .HI+〈L1,L2〉(d) =MI(1, d) : 0 0 1 1 1 1 . . .HI+〈L1〉(d) =MI(2, d) : 0 0 1 2 5 6 . . .
HI(d) =MI(3, d) : 0 0 1 3 8 14 . . .
The sectional matrix of P/I is
0 1 2 3 4 5 . . .HP/(I+〈L1,L2〉)(d) =MP/I(1, d) : 1 1 0 0 0 0 . . .
HP/(I+〈L1〉)(d) =MP/I(2, d) : 1 2 2 2 0 0 . . .
HP/I(d) =MP/I(3, d) : 1 3 5 7 7 7 . . .
Elisa Palezzato Ph.D. Seminar 6 March 2017 17 / 27
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Example
Let I be the zero-dimensional homogeneous ideal
(x2+y2−25z2, y4−3xy2z−4y3z+12xyz2−25y2z2+100yz3, xy3−16xyz2)
and rgin(I) =(x2, xy3, y4).
The sectional matrix of I is
0 1 2 3 4 5 . . .HI+〈L1,L2〉(d) =MI(1, d) : 0 0 1 1 1 1 . . .HI+〈L1〉(d) =MI(2, d) : 0 0 1 2 5 6 . . .
HI(d) =MI(3, d) : 0 0 1 3 8 14 . . .
The sectional matrix of P/I is
0 1 2 3 4 5 . . .HP/(I+〈L1,L2〉)(d) =MP/I(1, d) : 1 1 0 0 0 0 . . .
HP/(I+〈L1〉)(d) =MP/I(2, d) : 1 2 2 2 0 0 . . .
HP/I(d) =MP/I(3, d) : 1 3 5 7 7 7 . . .
Elisa Palezzato Ph.D. Seminar 6 March 2017 17 / 27
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Example
Let I be the zero-dimensional homogeneous ideal
(x2+y2−25z2, y4−3xy2z−4y3z+12xyz2−25y2z2+100yz3, xy3−16xyz2)
and rgin(I) =(x2, xy3, y4).
The sectional matrix of I is
0 1 2 3 4 5 . . .HI+〈L1,L2〉(d) =MI(1, d) : 0 0 1 1 1 1 . . .HI+〈L1〉(d) =MI(2, d) : 0 0 1 2 5 6 . . .
HI(d) =MI(3, d) : 0 0 1 3 8 14 . . .
The sectional matrix of P/I is
0 1 2 3 4 5 . . .HP/(I+〈L1,L2〉)(d) =MP/I(1, d) : 1 1 0 0 0 0 . . .
HP/(I+〈L1〉)(d) =MP/I(2, d) : 1 2 2 2 0 0 . . .
HP/I(d) =MP/I(3, d) : 1 3 5 7 7 7 . . .
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Additional Algebraic Properties I
Proposition (Bigatti-P.-Torielli)
Let I be a homogeneous ideal in P = K[x1, . . . , xn] and δ = reg(I).Suppose that
MP/I(i, δ) 6= 0 butMP/I(i− 1, δ) = 0 for some i = 2, . . . , n.
Then dim(P/I) = n− i+ 1 and deg(P/I) =MP/I(i, δ).
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Additional Algebraic Properties I
Proposition (Bigatti-P.-Torielli)
Let I be a homogeneous ideal in P = K[x1, . . . , xn] and δ = reg(I).Suppose that
MP/I(i, δ) 6= 0 butMP/I(i− 1, δ) = 0 for some i = 2, . . . , n.
Then dim(P/I) = n− i+ 1 and deg(P/I) =MP/I(i, δ).
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Additional Algebraic Properties II
Proposition (Bigatti-P.-Torielli)
Let I be a homogeneous ideal in P = K[x1, . . . , xn] such that the
minimal generators of I have degree ≤ δ. Suppose that
there exist i = 2, . . . , n and d ≥ δ such thatMP/I(k, d) = 0 for all
k = 1, . . . , i− 1;
MP/I(i, d) =MP/I(i, d+ 1) 6= 0.
Then dim(P/I) = n− i+ 1 and deg(P/I) =MP/I(i, d).
Elisa Palezzato Ph.D. Seminar 6 March 2017 19 / 27
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Additional Algebraic Properties II
Proposition (Bigatti-P.-Torielli)
Let I be a homogeneous ideal in P = K[x1, . . . , xn] such that the
minimal generators of I have degree ≤ δ. Suppose that
there exist i = 2, . . . , n and d ≥ δ such thatMP/I(k, d) = 0 for all
k = 1, . . . , i− 1;
MP/I(i, d) =MP/I(i, d+ 1) 6= 0.
Then dim(P/I) = n− i+ 1 and deg(P/I) =MP/I(i, d).
Elisa Palezzato Ph.D. Seminar 6 March 2017 19 / 27
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Additional Algebraic Properties II
Proposition (Bigatti-P.-Torielli)
Let I be a homogeneous ideal in P = K[x1, . . . , xn] such that the
minimal generators of I have degree ≤ δ. Suppose that
there exist i = 2, . . . , n and d ≥ δ such thatMP/I(k, d) = 0 for all
k = 1, . . . , i− 1;
MP/I(i, d) =MP/I(i, d+ 1) 6= 0.
Then dim(P/I) = n− i+ 1 and deg(P/I) =MP/I(i, d).
Elisa Palezzato Ph.D. Seminar 6 March 2017 19 / 27
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1 PREQUEL
2 Introduction
3 Sectional matrix and its algebraic properties
4 Geometrical properties
5 Examples
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Geometrical Properties I
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn] such that (I)δ 6= 0.Assume that one of the following holds
MP/I(2, δ) =MP/I(2, δ + 1) 6= 0.
MP/I(2, δ) 6= 0 andMP/I(n, δ + 1) =∑n
i=1MP/I(i, δ).
Then the ideals (I)≤δ and (I)≤δ+1 are saturated and their elements
have a GCD of degreeMP/I(2, δ).
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Geometrical Properties I
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn] such that (I)δ 6= 0.Assume that one of the following holds
MP/I(2, δ) =MP/I(2, δ + 1) 6= 0.
MP/I(2, δ) 6= 0 andMP/I(n, δ + 1) =∑n
i=1MP/I(i, δ).
Then the ideals (I)≤δ and (I)≤δ+1 are saturated and their elements
have a GCD of degreeMP/I(2, δ).
Elisa Palezzato Ph.D. Seminar 6 March 2017 21 / 27
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Geometrical Properties I
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn] such that (I)δ 6= 0.Assume that one of the following holds
MP/I(2, δ) =MP/I(2, δ + 1) 6= 0.
MP/I(2, δ) 6= 0 andMP/I(n, δ + 1) =∑n
i=1MP/I(i, δ).
Then the ideals (I)≤δ and (I)≤δ+1 are saturated and their elements
have a GCD of degreeMP/I(2, δ).
Elisa Palezzato Ph.D. Seminar 6 March 2017 21 / 27
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Geometrical Properties I
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn] such that (I)δ 6= 0.Assume that one of the following holds
MP/I(2, δ) =MP/I(2, δ + 1) 6= 0.
MP/I(2, δ) 6= 0 andMP/I(n, δ + 1) =∑n
i=1MP/I(i, δ).
Then the ideals (I)≤δ and (I)≤δ+1 are saturated and their elements
have a GCD of degreeMP/I(2, δ).
Elisa Palezzato Ph.D. Seminar 6 March 2017 21 / 27
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Geometrical Properties II
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn]. Suppose that exists δsuch that
0 =MP/I(1, δ) = · · · =MP/I(i− 1, δ);
MP/I(i, δ) 6= 0 for some n ≥ i ≥ 2;
MP/I(n, δ + 1) =∑n
k=iMP/I(k, δ).
Then the ideal (I)≤δ is a saturated, dim(P/(I)≤δ) = (n− i+ 1), and of
degreeMP/I(i, δ) and it is δ-regular. Moreover, dim(P/I) ≤ n− i+ 1.
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Geometrical Properties II
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn]. Suppose that exists δsuch that
0 =MP/I(1, δ) = · · · =MP/I(i− 1, δ);
MP/I(i, δ) 6= 0 for some n ≥ i ≥ 2;
MP/I(n, δ + 1) =∑n
k=iMP/I(k, δ).
Then the ideal (I)≤δ is a saturated, dim(P/(I)≤δ) = (n− i+ 1), and of
degreeMP/I(i, δ) and it is δ-regular. Moreover, dim(P/I) ≤ n− i+ 1.
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Geometrical Properties II
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn]. Suppose that exists δsuch that
0 =MP/I(1, δ) = · · · =MP/I(i− 1, δ);
MP/I(i, δ) 6= 0 for some n ≥ i ≥ 2;
MP/I(n, δ + 1) =∑n
k=iMP/I(k, δ).
Then the ideal (I)≤δ is a saturated, dim(P/(I)≤δ) = (n− i+ 1), and of
degreeMP/I(i, δ) and it is δ-regular. Moreover, dim(P/I) ≤ n− i+ 1.
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Geometrical Properties II
Theorem (Bigatti-P.-Torielli)
Let I be a saturated ideal of P = K[x1, . . . , xn]. Suppose that exists δsuch that
0 =MP/I(1, δ) = · · · =MP/I(i− 1, δ);
MP/I(i, δ) 6= 0 for some n ≥ i ≥ 2;
MP/I(n, δ + 1) =∑n
k=iMP/I(k, δ).
Then the ideal (I)≤δ is a saturated, dim(P/(I)≤δ) = (n− i+ 1), and of
degreeMP/I(i, δ) and it is δ-regular. Moreover, dim(P/I) ≤ n− i+ 1.
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1 PREQUEL
2 Introduction
3 Sectional matrix and its algebraic properties
4 Geometrical properties
5 Examples
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Example in P2
x
y
Consider the points
(0, 5), (0,−5), (5, 0), (−5, 0)(−3, 4), (3, 4), (−3,−4)
The de�ning ideal is I =
(x2+ y2− 25z2, y4+ . . . , . . . )
and rgin(I) =
(x2, xy3, y4).
The sectional matrix is
0 1 2 3 4 5 . . .1 1 0 0 0 0 . . .1 2 2 2 0 0 . . .1 3 5 7 7 7 . . .
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Example in P2
x
y
Consider the points
(0, 5), (0,−5), (5, 0), (−5, 0)(−3, 4), (3, 4), (−3,−4)
The de�ning ideal is I =
(x2+ y2− 25z2, y4+ . . . , . . . )
and rgin(I) =
(x2, xy3, y4).
The sectional matrix is
0 1 2 3 4 5 . . .1 1 0 0 0 0 . . .1 2 2 2 0 0 . . .1 3 5 7 7 7 . . .
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Example in P2
x
y
Consider the points
(0, 5), (0,−5), (5, 0), (−5, 0)(−3, 4), (3, 4), (−3,−4)
The de�ning ideal is I =
(x2+ y2− 25z2, y4+ . . . , . . . )
and rgin(I) =
(x2, xy3, y4).
The sectional matrix is
0 1 2 3 4 5 . . .1 1 0 0 0 0 . . .1 2 2 2 0 0 . . .1 3 5 7 7 7 . . .
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Example in P3
zx
yConsider the points
(0, 0, 0), (0, 1, 0), (0, 2, 0)
(0, 3, 0), (0, 4, 0), (0, 5, 0)
(2, 1, 0), (1, 1, 0), (0, 0, 3)
The de�ning ideal I issuch that rgin(I) =
(x2, xy, xz, y2, yz2, z6).
The sectional matrix is
0 1 2 3 4 5 6 . . .1 1 0 0 0 0 0 . . .1 2 0 0 0 0 0 . . .1 3 2 1 1 1 0 . . .1 4 6 7 8 9 9 . . .
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Example in P3
zx
yConsider the points
(0, 0, 0), (0, 1, 0), (0, 2, 0)
(0, 3, 0), (0, 4, 0), (0, 5, 0)
(2, 1, 0), (1, 1, 0), (0, 0, 3)
The de�ning ideal I issuch that rgin(I) =
(x2, xy, xz, y2, yz2, z6).
The sectional matrix is
0 1 2 3 4 5 6 . . .1 1 0 0 0 0 0 . . .1 2 0 0 0 0 0 . . .1 3 2 1 1 1 0 . . .1 4 6 7 8 9 9 . . .
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Example in P3
zx
yConsider the points
(0, 0, 0), (0, 1, 0), (0, 2, 0)
(0, 3, 0), (0, 4, 0), (0, 5, 0)
(2, 1, 0), (1, 1, 0), (0, 0, 3)
The de�ning ideal I issuch that rgin(I) =
(x2, xy, xz, y2, yz2, z6).
The sectional matrix is
0 1 2 3 4 5 6 . . .1 1 0 0 0 0 0 . . .1 2 0 0 0 0 0 . . .1 3 2 1 1 1 0 . . .1 4 6 7 8 9 9 . . .
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H-SM-gin-res
Hilbert function
sectional matrix
generic initial ideal (gin)
resolution
(Example)
Elisa Palezzato Ph.D. Seminar 6 March 2017 26 / 27
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H-SM-gin-res
Hilbert function
sectional matrix
generic initial ideal (gin)
resolution
(Example)
Elisa Palezzato Ph.D. Seminar 6 March 2017 26 / 27
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The End
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