95 research outputs found

### Betti tables forcing failure of the Weak Lefschetz Property

We study the Artinian reduction $A$ of a configuration of points $X \subset
{\mathbb P}^n$, and the relation of the geometry of $X$ to Lefschetz
properties of $A$. Migliore initiated the study of this connection, with a
particular focus on the Hilbert function of $A$, and further results appear in
work of Migliore--Mir\'o-Roig--Nagel. Our specific focus is on Betti tables
rather than Hilbert functions, and we prove that a certain type of Betti table
forces the failure of the Weak Lefschetz Property (WLP). The corresponding
Artinian algebras are typically not level, and the failure of WLP in these
cases is not detected in terms of the Hilbert function

### Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four

In 1978, Stanley constructed an example of an Artinian Gorenstein (AG) ring
$A$ with non-unimodal $H$-vector $(1,13,12,13,1)$. Migliore-Zanello later
showed that for regularity $r=4$, Stanley's example has the smallest possible
codimension $c$ for an AG ring with non-unimodal $H$-vector. The weak Lefschetz
property (WLP) has been much studied for AG rings; it is easy to show that an
AG ring with non-unimodal $H$-vector fails to have WLP. In codimension $c=3$ it
is conjectured that all AG rings have WLP. For $c=4$, Gondim showed that WLP
always holds for $r \le 4$ and gives a family where WLP fails for any $r \ge
7$, building on an earlier example of Ikeda of failure of WLP for $r=5$. In
this note we study the minimal free resolution of $A$ and relation to Lefschetz
properties (both weak and strong) and Jordan type for $c=4$ and $r \le 6$.Comment: 11 page

### Nets in $\mathbb P^2$ and Alexander Duality

A net in $\mathbb{P}^2$ is a configuration of lines $\mathcal A$ and points
$X$ satisfying certain incidence properties. Nets appear in a variety of
settings, ranging from quasigroups to combinatorial design to classification of
Kac-Moody algebras to cohomology jump loci of hyperplane arrangements. For a
matroid $M$ and rank $r$, we associate a monomial ideal (a monomial variant of
the Orlik-Solomon ideal) to the set of flats of $M$ of rank $\le r$. In the
context of line arrangements in $\mathbb{P}^2$, applying Alexander duality to
the resulting ideal yields insight into the combinatorial structure of nets.Comment: 15p

### The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in$\mathbb{P}^n$

International audienceFor a reduced hypersurface $V(f) \subseteq \mathbb{P}^n$ of degree $d$, theCastelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understoodwhen $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. Westudy the regularity of $M(f)$ when $V(f)$ has a positive dimensional singularlocus. In certain situations, we prove that the regularity is bounded by$(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However,this is not always the case, and we prove that in $\mathbb{P}^n$ the regularityof the Milnor algebra can grow quadratically in $d$

### The simplest minimal free resolutions in P1×P1

We study the minimal bigraded free resolution of an ideal with three generators of the same bidegree, contained in the bihomogeneous maximal ideal 〈s, t〉∩〈u, v〉 of the bigraded ring K[s,t;u,v]. Our analysis involves tools from algebraic geometry (Segre-Veronese varieties), classical commutative algebra (Buchsbaum-Eisenbud criteria for exactness, Hilbert-Burch theorem), and homological algebra (Koszul homology, spectral sequences). We treat in detail the case in which the bidegree is (1, n). We connect our work to a conjecture of Fröberg–Lundqvist on bigraded Hilbert functions, and close with a number of open problems.Fil: Botbol, Nicolas Santiago. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Dickenstein, Alicia Marcela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Schenck, Hal. Auburn University.; Estados Unido

### The Hessian polynomial and the Jacobian ideal of a reduced surface in$\mathbb{P}^3$

For a reduced hypersurface $V(f) \subseteq \mathbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study the regularity of $M(f)$ when $V(f)$ has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by $(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However, this is not always the case, and we prove that in $\mathbb{P}^3$ the regularity of the Milnor algebra can grow quadratically in $d$

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