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29 March 2024
 
  » arxiv » math.AC/0307281

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Ancestor ideals of vector spaces of forms, and level algebras
Anthony Iarrobino ;
Date 21 Jul 2003
Journal J. Algebra 272 (2004), no. 2, 530--580
Subject Commutative Algebra; Algebraic Geometry MSC-class: 13D40, 14C05 | math.AC math.AG
AbstractLet R be the polynomial ring in r variables over a field k, with maximal ideal M and let V denote a vector subspace of the space of degree-j homogeneous elements of R. We study three related algebras determined by V. The first is the ``ancestor algebra’’ whose defining ideal is the largest graded ideal whose intersection with M^j is the ideal (V). The second is the ``level algebra’’, whose defining ideal L(V) is the largest graded ideal of R such that the degree-j component is V; and third is the algebra R/(V). When r=2, we determine the possible Hilbert functions H for each of these algebras, and as well the dimension of each Hilbert function stratum. We characterize the graded Betti numbers of these algebras in terms of certain partitions depending only on H, and give the codimension of each stratum in terms of invariants of the partitions. When r=2 and k is algebraically closed the Hilbert function strata for each of the three algebras satisfy a frontier property that the closure of a stratum is the union of more special strata. The family G(H) of all graded quotients of R having the given Hilbert function is a natural desingularization of this closure.
Source arXiv, math.AC/0307281
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