Betti numbers of symmetric shifted ideals

Biermann, Jennifer; Alba, Hernán de; Galetto, Federico; Murai, Shinji; Nagel, Uwe; O'Keefe, Augustine B.; Römer, Tim; Seceleanu, Alexandra

doi:10.1016/j.jalgebra.2020.04.037

Cited by 20 publications

(16 citation statements)

References 24 publications

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“…At the time that this paper was being concluded, a preprint was posted on arXiv by J. Biermann, H. De Alba, F. Galetto, S. Murai, U. Nagel, A. O'Keefe, T. Römer and A. Seceleanu [8]. Independently from us, they prove result (4)(b) of Section 1.1 in the monomial case, i.e.…”

Section: Introduction Star Configuration Of Points In P Nmentioning

confidence: 78%

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The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces

Mantero¹

2019

Preprint

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Star configurations of hypersurfaces are schemes in P n widely generalizing star configurations of points. Their rich structure allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In particular, there has been much interest in understanding how "fattening" these schemes affects the algebraic properties of these configurations or, in other words, understanding the symbolic powers I (m) of their defining ideals I.In the present paper (1) we prove a structure theorem for I (m) , giving an explicit description of a minimal generating set of I (m) (overall, and in each degree) which also yields a minimal generating set of the module I (m) /I m -which measures how far is I (m) from I m . These results are new even for monomial star configurations or star configurations of points; (2) we introduce a notion of ideals with c.i. quotients, generalizing ideals with linear quotients, and show that I (m) have c.i. quotients. As a corollary we obtain that symbolic powers of ideals of star configurations of points have linear quotients; (3) we find a general formula for all graded Betti numbers of I (m) ; (4) we prove that a little bit more than the bottom half of the Betti table of I (m) has a regular, almost hypnotic, pattern, and provide a simple closed formula for all these graded Betti numbers and the last irregular strand in the Betti table.Other applications include improving and widely extending results by Galetto, Geramita, Shin and Van Tuyl, and providing explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations.Inspired by Young tableaux, we introduce a "canonical" way of writing any monomial in any given set of polynomials, which may be of independent interest. We prove its existence and uniqueness under fairly general assumption. Along the way, we exploit a connection between the minimal generators G (m) of I (m) and positive solutions to Diophantine equations, and a connection between G (m) and partitions of m via the canonical form of monomials. Our methods are characteristic-free.

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Section: Introduction Star Configuration Of Points In P Nmentioning

confidence: 78%

“…Independently from us, they prove result (4)(b) of Section 1.1 in the monomial case, i.e. when all forms have degree δ = 1 and are variables, see [8, Thm 3.2 and 4.3], (5) (see [8,Corollary 4.4(1)]) and a weaker version of (6) (see [8,Corollary 4.4] and Theorems 7.8 and 7.7 and Proposition 7.9).…”

Section: Introduction Star Configuration Of Points In P Nmentioning

confidence: 91%

The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces

Mantero¹

2019

Preprint

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“…Proof. Condition (1) always implies condition (2). Thus assume that I c,F satisfies the G s condition.…”

Section: The Linear Type Property Of Ideals Of Star Configurationsmentioning

confidence: 99%

“…From a commutative algebra perspective, ideals defining star configurations represent an interesting class, since a great amount of information is known about their free resolutions, Hilbert functions and symbolic powers (see for instance [13,14,11,22,24,2,3,28,23]). In this article we study their Rees algebras, about which little is currently known (see for instance [18,12,26,5]).…”

Section: Introductionmentioning

confidence: 99%

Rees algebras of ideals of star configurations

Costantini¹,

Drabkin²,

Guerrieri³

2021

Preprint

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“…∆ (4,1,1),(0,0) (I) = ∆ (5,2,0),(0,0) (I) = {∅}, ∆ (4,4,1),(1,0) (I) = ∆ (5,5,0), (1) (I) = {∅}, ∆ (5,2,1),(0,0,0) = {1, 2}, {1}, {2}, {3}, ∅ , ∆ (4,4,4), (2) (I) = {∅} and ∆ (5,5,1),(1,0) = {1}, {2}, ∅ .…”

Section: Introductionmentioning

confidence: 99%

An equivariant Hochster's formula for $\mathfrak S_n$-invariant monomial ideals

Murai,

Raicu

2020

Preprint

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Let R = k[x 1 , . . . , x n ] be a polynomial ring over a field k and let I ⊂ R be a monomial ideal preserved by the natural action of the symmetric group S n on R. We give a combinatorial method to determine the S n -module structure of Tor i (I, k). Our formula shows that Tor i (I, k) is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an S n -equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of S n -invariant monomial ideals, and in particular recover formulas for their Castelnuovo-Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or > n) we compute the S n -invariant part of Tor i (I, k) in terms of Tor groups of the unsymmetrization of I.

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Betti numbers of symmetric shifted ideals

Cited by 20 publications

References 24 publications

The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces

The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces

Rees algebras of ideals of star configurations

An equivariant Hochster's formula for $\mathfrak S_n$-invariant monomial ideals

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