Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0307281

Ancestor ideals of vector spaces of forms, and level algebras

Anthony Iarrobino
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let 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.

Explore related subjects

math.ACmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Anthony Iarrobino. 2003-07-21. Ancestor ideals of vector spaces of forms, and level algebras. https://arxiv.org/abs/math/0307281

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action

T. Kambayashi had shown that $\mathbb{A}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable $\mathbb{A}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for $\mathbb{A}^2$-forms over arbitrary field extensions admitting a nontrivial $\mathbb{G}_a$-action. From this structure theorem we derive some conditions under which an $\mathbb{A}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial $\mathbb{A}^2$-form having a $k$-rational point and a non-trivial $\mathbb{G}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field.

math.AC↗
arXiv2026

Symbolic powers of the ideal of$n$ general points in $P^{n-1}$

Problem L of Fröberg--Lundqvist--Oneto--Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for \(n\) general points of \(\PP^{n-1}\). Besides a closed formula for \[ \HS(S/I^m)-\HS(S/I^{(m)}), \] we determine all minimal monomial generators of \(I^{(m)}\), and describe the symbolic Rees algebra. We also show that containment \(I^{(m)}\subseteq I^r\) is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant \(\walpha\), the resurgence \(\Res\), and the asymptotic resurgence \(\aRes\): \[ \walpha(I)=\frac{n}{n-1}, \qquad \Res(I)=\aRes(I)=\frac{2(n-1)}{n}. \] We also take the first step beyond \(n\) points: for \(n+1\) general points of \(\PP^{n-1}\) --- again a rigid, non-monomial configuration --- we identify the defining quadrics, resolve the case \(n=3\) completely (a complete intersection, with \(J^{(m)}=J^m\) for all \(m\) and resurgence \(1\)), and propose an exact Waldschmidt-constant formula \(\walpha=\frac{n+1}{n-1}\) for all \(n\), verified computationally in every case we could check.

math.AC↗