Ph.D. Seminar, University of Genoav - DIMA · 2017. 4. 6. · Extremal Behaviour in Sectional...

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

Elisa Palezzato Ph.D. Seminar 6 March 2017 1 / 27

Extremal Behaviour in Sectional Matrices

1 PREQUEL

2 Introduction

3 Sectional matrix and its algebraic properties

4 Geometrical properties

5 Examples

1 PREQUEL

2 Introduction

5 Examples

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, ...]

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

1 PREQUEL

2 Introduction

5 Examples

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.

Hilbert Function

De�nition

HI(d) := dimK(Id).

Hilbert Function

De�nition

HI(d) := dimK(Id).

Hilbert Function

De�nition

HI(d) := dimK(Id).

Question

Can the Hilbert function characterize also some of the geometrical

behaviour of algebraic variety in the projective space?

Answer

In general no.

Question

Can the Hilbert function characterize also some of the geometrical

behaviour of algebraic variety in the projective space?

Answer

In general no.

Example in P2

Example in P3

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.

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.

1 PREQUEL

2 Introduction

5 Examples

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)

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)

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.

De�nition

Example

Remark

De�nition

Example

Remark

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).

De�nition

Example

De�nition

Example

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.

Proposition

generators of degree ≤ δ. Then1 MI(n,−) coincides with the Hilbert function of I, andMP/I(n,−)

with the one of P/I.

3 MP/I(k, d+ 1) ≤∑k

MP/I(k, δ + 1) =∑k

Proposition

3 MP/I(k, d+ 1) ≤∑k

MP/I(k, δ + 1) =∑k

Proposition

3 MP/I(k, d+ 1) ≤∑k

MP/I(k, δ + 1) =∑k

Proposition

3 MP/I(k, d+ 1) ≤∑k

MP/I(k, δ + 1) =∑k

Proposition

3 MP/I(k, d+ 1) ≤∑k

MP/I(k, δ + 1) =∑k

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 . . .

Example

HI(d) =MI(3, d) : 0 0 1 3 8 14 . . .

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 . . .

Example

HI(d) =MI(3, d) : 0 0 1 3 8 14 . . .

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 . . .

Example

HI(d) =MI(3, d) : 0 0 1 3 8 14 . . .

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 . . .

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).

Additional Algebraic Properties I

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).

Additional Algebraic Properties II

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).

k = 1, . . . , i− 1;

MP/I(i, d) =MP/I(i, d+ 1) 6= 0.

k = 1, . . . , i− 1;

MP/I(i, d) =MP/I(i, d+ 1) 6= 0.

1 PREQUEL

2 Introduction

5 Examples

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).

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).

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).

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).

Geometrical Properties II

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.

0 =MP/I(1, δ) = · · · =MP/I(i− 1, δ);

MP/I(n, δ + 1) =∑n

k=iMP/I(k, δ).

0 =MP/I(1, δ) = · · · =MP/I(i− 1, δ);

MP/I(n, δ + 1) =∑n

k=iMP/I(k, δ).

0 =MP/I(1, δ) = · · · =MP/I(i− 1, δ);

MP/I(n, δ + 1) =∑n

k=iMP/I(k, δ).

1 PREQUEL

2 Introduction

5 Examples

Example in P2

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 . . .

Example in P2

Consider the points

(0, 5), (0,−5), (5, 0), (−5, 0)(−3, 4), (3, 4), (−3,−4)

(x2+ y2− 25z2, y4+ . . . , . . . )

and rgin(I) =

(x2, xy3, y4).

0 1 2 3 4 5 . . .1 1 0 0 0 0 . . .1 2 2 2 0 0 . . .1 3 5 7 7 7 . . .

Example in P2

Consider the points

(0, 5), (0,−5), (5, 0), (−5, 0)(−3, 4), (3, 4), (−3,−4)

(x2+ y2− 25z2, y4+ . . . , . . . )

and rgin(I) =

(x2, xy3, y4).

0 1 2 3 4 5 . . .1 1 0 0 0 0 . . .1 2 2 2 0 0 . . .1 3 5 7 7 7 . . .

Example in P3

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).

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 . . .

Example in P3

(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)

(x2, xy, xz, y2, yz2, z6).

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 . . .

Example in P3

(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)

(x2, xy, xz, y2, yz2, z6).

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 . . .

H-SM-gin-res

Hilbert function

sectional matrix

generic initial ideal (gin)

resolution

(Example)

H-SM-gin-res

Hilbert function

sectional matrix

generic initial ideal (gin)

resolution

(Example)

The End

Ph.D. Seminar, University of Genoav - DIMA · 2017. 4. 6. · Extremal Behaviour in Sectional...

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