95 research outputs found

    Betti tables forcing failure of the Weak Lefschetz Property

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    We study the Artinian reduction AA of a configuration of points X⊂PnX \subset {\mathbb P}^n , and the relation of the geometry of XX to Lefschetz properties of AA. Migliore initiated the study of this connection, with a particular focus on the Hilbert function of AA, 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

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

    Nets in P2\mathbb P^2 and Alexander Duality

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    A net in P2\mathbb{P}^2 is a configuration of lines A\mathcal A and points XX 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 MM and rank rr, we associate a monomial ideal (a monomial variant of the Orlik-Solomon ideal) to the set of flats of MM of rank ≤r\le r. In the context of line arrangements in P2\mathbb{P}^2, applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.Comment: 15p

    Computational algebraic geometry

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    xiv, 193 p. : ill. ; 24 cm

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

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

    The simplest minimal free resolutions in P1×P1

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    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 inP3\mathbb{P}^3

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    For a reduced hypersurface V(f)⊆PnV(f) \subseteq \mathbb{P}^n of degree dd, the Castelnuovo-Mumford regularity of the Milnor algebra M(f)M(f) is well understood when V(f)V(f) is smooth, as well as when V(f)V(f) has isolated singularities. We study the regularity of M(f)M(f) when V(f)V(f) has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by (d−2)(n+1)(d-2)(n+1), which is the degree of the Hessian polynomial of ff. However, this is not always the case, and we prove that in P3\mathbb{P}^3 the regularity of the Milnor algebra can grow quadratically in dd
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