arXiv subjects

Whitney Liske

Publications and source records attributed to Whitney Liske.

3 recordsLinked to original sources

arXiv2025

Algebraic invariants of the special fiber ring of ladder determinantal modules

We provide explicit formulas for key invariants of special fiber rings of ladder determinantal modules, that is, modules that are direct sums of ideals of maximal minors of a ladder matrix. Our results are given in terms of the combinatorial data of the associated ladder matrix. In particular, we compute its dimension, regularity, $a$-invariant, and multiplicity, which via \textsc{Sagbi} degeneration coincide with those of Hibi rings associated to a distributive lattice. Then, via Gr\"{o}bner degeneration these calculations are reduced to quotients of polynomial rings by monomial ideals. Our formula for the multiplicity of the special fiber ring of these ladder determinantal modules is obtained by counting the number of standard skew Young tableaux associated to a certain skew partition, and so provides a natural generalization of the classical formula for the degree of the Grassmannian.

math.AC↗

arXiv2024

Rees algebras of ideals submaximally generated by quadrics

The goal of this paper is to study the Rees algebra $\mathfrak{R}(I)$and the special fiber ring $\mathfrak{F}(I)$ for a family of ideals. Let $R=\mathbb{K}[x_1, \ldots, x_d]$ with $d\geq 4$ be a polynomial ring with homogeneous maximal ideal $\mathfrak{m}$. We study the $R$-ideals $I$, which are $\mathfrak{m}$-primary, Gorenstein, generated in degree 2, and have a Gorenstein linear resolution. In the smallest case, $d=4$, this family includes the ideals of $2\times 2$ minors of a general $3\times 3$ matrix of linear forms in $R$. We show that the defining ideal of the Rees algebra will be of fiber type. That is, the defining ideal of the Rees algebra is generated by the defining ideals of the special fiber ring and of the symmetric algebra. We use the fact that these ideals differ from $\mathfrak{m}^2$ by exactly one minimal generator to describe the defining ideal $\mathfrak{F}(I)$ as a sub-ideal of the defining ideal of $\mathfrak{F}(\mathfrak{m}^2)$, which is well known to be the ideal of $2\times 2$ minors of a symmetric matrix of variables.

math.AC↗

arXiv2023

Rees Algebras of Unit Interval Determinantal Facet Ideals

Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebras and special fiber rings of unit interval determinantal facet ideals. In particular, we show that unit interval determinantal facet ideals are of fiber type and that their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains and have rational singularities.

math.AC↗