Extremal Behaviour in Sectional Matrices

Transcription

1 Extremal Behaviour in Sectional Matrices Elisa Palezzato 1 joint work with Anna Maria Bigatti 1 and Michele Torielli 2 1 University of Genova, Italy 2 Hokkaido University, Japan arxiv: Ph.D. Seminar, University of Genova 6 March 2017 Elisa Palezzato Ph.D. Seminar 6 March / 27

2 Extremal Behaviour in Sectional Matrices 1 PREQUEL 2 Introduction 3 Sectional matrix and its algebraic properties 4 Geometrical properties 5 Examples Elisa Palezzato Ph.D. Seminar 6 March / 27

3 1 PREQUEL 2 Introduction 3 Sectional matrix and its algebraic properties 4 Geometrical properties 5 Examples Elisa Palezzato Ph.D. Seminar 6 March / 27

4 From Wikipedia I A Computer Algebra System is a software program that allows computation over mathematical expressions in a way which is similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems started in the second half 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 specic part of mathematics, such as number theory, group theory, etc. [CoCoA, Macaulay2, Singular,...] General purpose computer algebra systems aim to be useful to a user working in any scientic eld that requires manipulation of mathematical expressions. [Matlab, Maple, Magma,...] Elisa Palezzato Ph.D. Seminar 6 March / 27

5 From Wikipedia II A Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a eld K[x 1,..., x n ]. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclidian algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. What you need to compute a Gröbner Basis: polynomial ring monomial ordering reduction algorithm Elisa Palezzato Ph.D. Seminar 6 March / 27

6 1 PREQUEL 2 Introduction 3 Sectional matrix and its algebraic properties 4 Geometrical properties 5 Examples Elisa Palezzato Ph.D. Seminar 6 March / 27

7 Hilbert Function The computation of Hilbert function is available in most computer algebra systems. Elisa Palezzato Ph.D. Seminar 6 March / 27

8 Hilbert Function The computation of Hilbert function is available in most computer algebra systems. Denition Let K be a eld of characteristic 0. Given a homogeneous ideal I in P = K[x 1,..., x n ], we dene the Hilbert function of I to be the function H I (d) := dim K (I d ). Elisa Palezzato Ph.D. Seminar 6 March / 27

9 Hilbert Function The computation of Hilbert function is available in most computer algebra systems. Denition Let K be a eld of characteristic 0. Given a homogeneous ideal I in P = K[x 1,..., x n ], we dene the Hilbert function of I to be the function H I (d) := dim K (I d ). Similarly, we dene the Hilbert function of P/I to be the function H P/I (d) := dim K ((P/I) d ). Elisa Palezzato Ph.D. Seminar 6 March / 27

10 Hilbert Function The computation of Hilbert function is available in most computer algebra systems. Denition Let K be a eld of characteristic 0. Given a homogeneous ideal I in P = K[x 1,..., x n ], we dene the Hilbert function of I to be the function H I (d) := dim K (I d ). Similarly, we dene the Hilbert function of P/I to be the function H P/I (d) := dim K ((P/I) d ). The Hilbert function is important in computational algebraic geometry, as it is the easiest known way for computing the dimension and the degree of an algebraic variety dened by explicit polynomial equations. Elisa Palezzato Ph.D. Seminar 6 March / 27

11 Question Can the Hilbert function characterize also some of the geometrical behaviour of algebraic variety in the projective space? Elisa Palezzato Ph.D. Seminar 6 March / 27

12 Question Can the Hilbert function characterize also some of the geometrical behaviour of algebraic variety in the projective space? Answer In general no. Elisa Palezzato Ph.D. Seminar 6 March / 27

13 Example in P 2 x y Elisa Palezzato Ph.D. Seminar 6 March / 27

14 Example in P 3 y z x Elisa Palezzato Ph.D. Seminar 6 March / 27

15 Question Can we nd an other algebraic invariant that characterize some of the geometrical behaviour of algebraic variety in the projective space? Elisa Palezzato Ph.D. Seminar 6 March / 27

16 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. Elisa Palezzato Ph.D. Seminar 6 March / 27

17 1 PREQUEL 2 Introduction 3 Sectional matrix and its algebraic properties 4 Geometrical properties 5 Examples Elisa Palezzato Ph.D. Seminar 6 March / 27

18 Sectional Matrix Denition Let K be a eld of characteristic 0. Given a homogeneous ideal I in P = K[x 1,..., x n ], we dene the sectional matrix of I to be the function M I (i, d) := dim K (I + (L 1,..., L n i )/(L 1,..., L n i )) d where L 1,..., L n i are general linear forms, i = 1,..., n and d 0. Elisa Palezzato Ph.D. Seminar 6 March / 27

19 Sectional Matrix Denition Let K be a eld of characteristic 0. Given a homogeneous ideal I in P = K[x 1,..., x n ], we dene the sectional matrix of I to be the function M I (i, d) := dim K (I + (L 1,..., L n i )/(L 1,..., L n i )) d where L 1,..., L n i are general linear forms, i = 1,..., n and d 0. Similarly, we dene the sectional matrix of P/I to be the function M P/I (i, d) := dim K (P/(I + (L 1,..., L n i )/(L 1,..., L n i ))) d ( ) d + i 1 = M I (i, d) i 1 where L 1,..., L n i are general linear forms, i = 1,..., n and d 0. Elisa Palezzato Ph.D. Seminar 6 March / 27

20 Strongly Stable Ideal Denition Let I be a homogenous ideal in P = K[x 1,..., x n ]. We say that I is a strongly stable ideal if T = x i 1 1 x in n I, then x i T/x j I for all i < j max{k i k 0}. Elisa Palezzato Ph.D. Seminar 6 March / 27

21 Strongly Stable Ideal Denition Let I be a homogenous ideal in P = K[x 1,..., x n ]. We say that I is a strongly stable ideal if T = x i 1 1 x in n I, then x i T/x j I for all i < j max{k i k 0}. Example The ideal I = (x 3, x 2 y, xy 2, xyz) is not a strongly stable in Q[x, y, z] because x xyz/y = x 2 z I. The ideal I + (x 2 z) is strongly stable. Elisa Palezzato Ph.D. Seminar 6 March / 27

22 Strongly Stable Ideal Denition Let I be a homogenous ideal in P = K[x 1,..., x n ]. We say that I is a strongly stable ideal if T = x i 1 1 x in n I, then x i T/x j I for all i < j max{k i k 0}. Example The ideal I = (x 3, x 2 y, xy 2, xyz) is not a strongly stable in Q[x, y, z] because x xyz/y = x 2 z I. The ideal I + (x 2 z) is strongly stable. Remark If I is a strongly stable ideal, then in the denition of the sectional matrix we can take L i = x n i+1. Elisa Palezzato Ph.D. Seminar 6 March / 27

23 Generic Initial Ideal Theorem (Galligo '74) Let I be a homogeneous ideal in K[x 1,..., x n ], σ a term-ordering such that x 1 > σ x 2 > σ > σ x n. Then there exists a Zariski open set U GL(n) and a strongly stable ideal J such that for each g U, LT σ (g(i)) = J. Elisa Palezzato Ph.D. Seminar 6 March / 27

24 Generic Initial Ideal Theorem (Galligo '74) Let I be a homogeneous ideal in K[x 1,..., x n ], σ a term-ordering such that x 1 > σ x 2 > σ > σ x n. Then there exists a Zariski open set U GL(n) and a strongly stable ideal J such that for each g U, LT σ (g(i)) = J. Denition The strongly stable ideal J given in the previous Theorem will be called the generic initial ideal with respect to σ of I and it will be denoted by gin σ (I). In particular, gin DegRevLex (I) is denoted by many authors with rgin(i). Elisa Palezzato Ph.D. Seminar 6 March / 27

25 Generic Initial Ideal Theorem (Galligo '74) Let I be a homogeneous ideal in K[x 1,..., x n ], σ a term-ordering such that x 1 > σ x 2 > σ > σ x n. Then there exists a Zariski open set U GL(n) and a strongly stable ideal J such that for each g U, LT σ (g(i)) = J. Denition The strongly stable ideal J given in the previous Theorem will be called the generic initial ideal with respect to σ of I and it will be denoted by gin σ (I). In particular, gin DegRevLex (I) is denoted by many authors with rgin(i). Example Consider the ideal I = (z 5, xyz 3 ) in Q[x, y, z], then rgin(i) = (x 5, x 4 y, x 3 y 3 ). Elisa Palezzato Ph.D. Seminar 6 March / 27

26 Properties of the Sectional Matrix Proposition Let I be a homogeneous ideal in P = K[x 1,..., x n ] with minimal generators of degree δ. Then Elisa Palezzato Ph.D. Seminar 6 March / 27

27 Properties of the Sectional Matrix Proposition Let I be a homogeneous ideal in P = K[x 1,..., x n ] with minimal generators of degree δ. Then 1 M I (n, ) coincides with the Hilbert function of I, and M P/I (n, ) with the one of P/I. Elisa Palezzato Ph.D. Seminar 6 March / 27

28 Properties of the Sectional Matrix Proposition Let I be a homogeneous ideal in P = K[x 1,..., x n ] with minimal generators of degree δ. Then 1 M I (n, ) coincides with the Hilbert function of I, and M P/I (n, ) with the one of P/I. 2 M I = M rgin(i), and M P/I = M P/rgin(I). Elisa Palezzato Ph.D. Seminar 6 March / 27

29 Properties of the Sectional Matrix Proposition Let I be a homogeneous ideal in P = K[x 1,..., x n ] with minimal generators of degree δ. Then 1 M I (n, ) coincides with the Hilbert function of I, and M P/I (n, ) with the one of P/I. 2 M I = M rgin(i), and M P/I = M P/rgin(I). 3 M P/I (k, d + 1) k i=1 M P/I(i, d), for all k and d. If we have M P/I (k, δ + 1) = k i=1 M P/I(i, δ), then M P/I (s, d + 1) = s i=1 M P/I(i, d), for all s k and d δ. Elisa Palezzato Ph.D. Seminar 6 March / 27

30 Properties of the Sectional Matrix Proposition Let I be a homogeneous ideal in P = K[x 1,..., x n ] with minimal generators of degree δ. Then 1 M I (n, ) coincides with the Hilbert function of I, and M P/I (n, ) with the one of P/I. 2 M I = M rgin(i), and M P/I = M P/rgin(I). 3 M P/I (k, d + 1) k i=1 M P/I(i, d), for all k and d. If we have M P/I (k, δ + 1) = k i=1 M P/I(i, δ), then M P/I (s, d + 1) = s i=1 M P/I(i, d), for all s k and d δ. 4 If I is a strongly stable ideal (I = rgin(i)), then M P/I (k, d + 1) = k i=1 M P/I(i, d), for all d > δ and for all k. Elisa Palezzato Ph.D. Seminar 6 March / 27

31 Properties of the Sectional Matrix Proposition Let I be a homogeneous ideal in P = K[x 1,..., x n ] with minimal generators of degree δ. Then 1 M I (n, ) coincides with the Hilbert function of I, and M P/I (n, ) with the one of P/I. 2 M I = M rgin(i), and M P/I = M P/rgin(I). 3 M P/I (k, d + 1) k i=1 M P/I(i, d), for all k and d. If we have M P/I (k, δ + 1) = k i=1 M P/I(i, δ), then M P/I (s, d + 1) = s i=1 M P/I(i, d), for all s k and d δ. 4 If I is a strongly stable ideal (I = rgin(i)), then M P/I (k, d + 1) = k i=1 M P/I(i, d), for all d > δ and for all k. 5 If δ = reg(i), then M P/I (k, d + 1) = k i=1 M P/I(i, d), for all d > δ and for all k. Elisa Palezzato Ph.D. Seminar 6 March / 27

32 Example Let I be the zero-dimensional homogeneous ideal (x 2 +y 2 25z 2, y 4 3xy 2 z 4y 3 z+12xyz 2 25y 2 z yz 3, xy 3 16xyz 2 ) Elisa Palezzato Ph.D. Seminar 6 March / 27

33 Example Let I be the zero-dimensional homogeneous ideal (x 2 +y 2 25z 2, y 4 3xy 2 z 4y 3 z+12xyz 2 25y 2 z yz 3, xy 3 16xyz 2 ) and rgin(i) = (x 2, xy 3, y 4 ). Elisa Palezzato Ph.D. Seminar 6 March / 27

34 Example Let I be the zero-dimensional homogeneous ideal (x 2 +y 2 25z 2, y 4 3xy 2 z 4y 3 z+12xyz 2 25y 2 z yz 3, xy 3 16xyz 2 ) and rgin(i) = The sectional matrix of I is (x 2, xy 3, y 4 ) H I+ L1,L 2 (d) = M I (1, d) : H I+ L1 (d) = M I (2, d) : H I (d) = M I (3, d) : Elisa Palezzato Ph.D. Seminar 6 March / 27

35 Example Let I be the zero-dimensional homogeneous ideal (x 2 +y 2 25z 2, y 4 3xy 2 z 4y 3 z+12xyz 2 25y 2 z yz 3, xy 3 16xyz 2 ) and rgin(i) = The sectional matrix of I is (x 2, xy 3, y 4 ) H I+ L1,L 2 (d) = M I (1, d) : H I+ L1 (d) = M I (2, d) : H I (d) = M I (3, d) : The sectional matrix of P/I is H P/(I+ L1,L 2 )(d) = M P/I (1, d) : H P/(I+ L1 )(d) = M P/I (2, d) : H P/I (d) = M P/I (3, d) : Elisa Palezzato Ph.D. Seminar 6 March / 27

36 Additional Algebraic Properties I Proposition (Bigatti-P.-Torielli) Let I be a homogeneous ideal in P = K[x 1,..., x n ] and δ = reg(i). Suppose that M P/I (i, δ) 0 but M P/I (i 1, δ) = 0 for some i = 2,..., n. Elisa Palezzato Ph.D. Seminar 6 March / 27

37 Additional Algebraic Properties I Proposition (Bigatti-P.-Torielli) Let I be a homogeneous ideal in P = K[x 1,..., x n ] and δ = reg(i). Suppose that M P/I (i, δ) 0 but M P/I (i 1, δ) = 0 for some i = 2,..., n. Then dim(p/i) = n i + 1 and deg(p/i) = M P/I (i, δ). Elisa Palezzato Ph.D. Seminar 6 March / 27

38 Additional Algebraic Properties II Proposition (Bigatti-P.-Torielli) Let I be a homogeneous ideal in P = K[x 1,..., x n ] such that the minimal generators of I have degree δ. Suppose that there exist i = 2,..., n and d δ such that M P/I (k, d) = 0 for all k = 1,..., i 1; Elisa Palezzato Ph.D. Seminar 6 March / 27

39 Additional Algebraic Properties II Proposition (Bigatti-P.-Torielli) Let I be a homogeneous ideal in P = K[x 1,..., x n ] such that the minimal generators of I have degree δ. Suppose that there exist i = 2,..., n and d δ such that M P/I (k, d) = 0 for all k = 1,..., i 1; M P/I (i, d) = M P/I (i, d + 1) 0. Elisa Palezzato Ph.D. Seminar 6 March / 27

40 Additional Algebraic Properties II Proposition (Bigatti-P.-Torielli) Let I be a homogeneous ideal in P = K[x 1,..., x n ] such that the minimal generators of I have degree δ. Suppose that there exist i = 2,..., n and d δ such that M P/I (k, d) = 0 for all k = 1,..., i 1; M P/I (i, d) = M P/I (i, d + 1) 0. Then dim(p/i) = n i + 1 and deg(p/i) = M P/I (i, d). Elisa Palezzato Ph.D. Seminar 6 March / 27

41 1 PREQUEL 2 Introduction 3 Sectional matrix and its algebraic properties 4 Geometrical properties 5 Examples Elisa Palezzato Ph.D. Seminar 6 March / 27

42 Geometrical Properties I Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ] such that (I) δ 0. Assume that one of the following holds Elisa Palezzato Ph.D. Seminar 6 March / 27

43 Geometrical Properties I Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ] such that (I) δ 0. Assume that one of the following holds M P/I (2, δ) = M P/I (2, δ + 1) 0. Elisa Palezzato Ph.D. Seminar 6 March / 27

44 Geometrical Properties I Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ] such that (I) δ 0. Assume that one of the following holds M P/I (2, δ) = M P/I (2, δ + 1) 0. M P/I (2, δ) 0 and M P/I (n, δ + 1) = n i=1 M P/I(i, δ). Elisa Palezzato Ph.D. Seminar 6 March / 27

45 Geometrical Properties I Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ] such that (I) δ 0. Assume that one of the following holds M P/I (2, δ) = M P/I (2, δ + 1) 0. M P/I (2, δ) 0 and M P/I (n, δ + 1) = n i=1 M P/I(i, δ). Then the ideals (I) δ and (I) δ+1 are saturated and their elements have a GCD of degree M P/I (2, δ). Elisa Palezzato Ph.D. Seminar 6 March / 27

46 Geometrical Properties II Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ]. Suppose that exists δ such that 0 = M P/I (1, δ) = = M P/I (i 1, δ); Elisa Palezzato Ph.D. Seminar 6 March / 27

47 Geometrical Properties II Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ]. Suppose that exists δ such that 0 = M P/I (1, δ) = = M P/I (i 1, δ); M P/I (i, δ) 0 for some n i 2; Elisa Palezzato Ph.D. Seminar 6 March / 27

48 Geometrical Properties II Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ]. Suppose that exists δ such that 0 = M P/I (1, δ) = = M P/I (i 1, δ); M P/I (i, δ) 0 for some n i 2; M P/I (n, δ + 1) = n k=i M P/I(k, δ). Elisa Palezzato Ph.D. Seminar 6 March / 27

49 Geometrical Properties II Theorem (Bigatti-P.-Torielli) Let I be a saturated ideal of P = K[x 1,..., x n ]. Suppose that exists δ such that 0 = M P/I (1, δ) = = M P/I (i 1, δ); M P/I (i, δ) 0 for some n i 2; M P/I (n, δ + 1) = n k=i M P/I(k, δ). Then the ideal (I) δ is a saturated, dim(p/(i) δ ) = (n i + 1), and of degree M P/I (i, δ) and it is δ-regular. Moreover, dim(p/i) n i + 1. Elisa Palezzato Ph.D. Seminar 6 March / 27

50 1 PREQUEL 2 Introduction 3 Sectional matrix and its algebraic properties 4 Geometrical properties 5 Examples Elisa Palezzato Ph.D. Seminar 6 March / 27

51 Example in P 2 x Consider the points (0, 5), (0, 5), (5, 0), ( 5, 0) ( 3, 4), (3, 4), ( 3, 4) y Elisa Palezzato Ph.D. Seminar 6 March / 27

52 Example in P 2 x Consider the points (0, 5), (0, 5), (5, 0), ( 5, 0) ( 3, 4), (3, 4), ( 3, 4) The dening ideal is I = (x 2 + y 2 25z 2, y ,... ) y and rgin(i) = (x 2, xy 3, y 4 ). Elisa Palezzato Ph.D. Seminar 6 March / 27

53 Example in P 2 x Consider the points (0, 5), (0, 5), (5, 0), ( 5, 0) ( 3, 4), (3, 4), ( 3, 4) The dening ideal is I = (x 2 + y 2 25z 2, y ,... ) y and rgin(i) = (x 2, xy 3, y 4 ). The sectional matrix is Elisa Palezzato Ph.D. Seminar 6 March / 27

54 Example in P 3 y Consider 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) z x Elisa Palezzato Ph.D. Seminar 6 March / 27

55 Example in P 3 y Consider 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 dening ideal I is such that rgin(i) = (x 2, xy, xz, y 2, yz 2, z 6 ). z x Elisa Palezzato Ph.D. Seminar 6 March / 27

56 Example in P 3 y Consider 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) z x The dening ideal I is such that rgin(i) = (x 2, xy, xz, y 2, yz 2, z 6 ). The sectional matrix is Elisa Palezzato Ph.D. Seminar 6 March / 27

57 H-SM-gin-res Hilbert function sectional matrix generic initial ideal (gin) resolution (Example) Elisa Palezzato Ph.D. Seminar 6 March / 27

58 H-SM-gin-res Hilbert function sectional matrix generic initial ideal (gin) resolution (Example) Elisa Palezzato Ph.D. Seminar 6 March / 27

59 The End Elisa Palezzato Ph.D. Seminar 6 March / 27