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Ladder determinantal varieties and their symbolic blowups

Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt, Lisa Seccia, Matteo Varbaro

TL;DR

This work establishes strong $F$-regularity for the symbolic Rees algebras and $F$-purity for the symbolic associated graded rings of two-sided mixed ladder determinantal ideals, deepening understanding of symbolic blowups in positive characteristic. It introduces chamfering as a crucial combinatorial tool to reduce mixed ladders to tractable configurations and to prove Noetherianity and compatibility with initial degenerations. The authors also connect these results to Knutson ideals, showing that several determinantal families, including Schubert and poset ideals, are Knutson and hence $F$-pure under positive characteristic, with squarefree initial ideals under antidiagonal term orders. Overall, the paper links ladder determinantal geometry with $F$-singularity theory, Gröbner degenerations, and algebras with straightening laws, yielding new structural and computational consequences for symbolic blowups and related algebras.

Abstract

In this article we show that the symbolic Rees algebra of a mixed ladder determinantal ideal is strongly $F$-regular. Furthermore, we prove that the symbolic associated graded algebra of a mixed ladder determinantal ideal is $F$-pure. The latter implies that mixed ladder determinantal rings are $F$-pure. We also show that ideals of the poset of minors of a generic matrix give rise to $F$-pure algebras with straightening law.

Ladder determinantal varieties and their symbolic blowups

TL;DR

This work establishes strong -regularity for the symbolic Rees algebras and -purity for the symbolic associated graded rings of two-sided mixed ladder determinantal ideals, deepening understanding of symbolic blowups in positive characteristic. It introduces chamfering as a crucial combinatorial tool to reduce mixed ladders to tractable configurations and to prove Noetherianity and compatibility with initial degenerations. The authors also connect these results to Knutson ideals, showing that several determinantal families, including Schubert and poset ideals, are Knutson and hence -pure under positive characteristic, with squarefree initial ideals under antidiagonal term orders. Overall, the paper links ladder determinantal geometry with -singularity theory, Gröbner degenerations, and algebras with straightening laws, yielding new structural and computational consequences for symbolic blowups and related algebras.

Abstract

In this article we show that the symbolic Rees algebra of a mixed ladder determinantal ideal is strongly -regular. Furthermore, we prove that the symbolic associated graded algebra of a mixed ladder determinantal ideal is -pure. The latter implies that mixed ladder determinantal rings are -pure. We also show that ideals of the poset of minors of a generic matrix give rise to -pure algebras with straightening law.
Paper Structure (9 sections, 19 theorems, 50 equations, 2 figures)

This paper contains 9 sections, 19 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Methods in prime characteristic
  4. Knutson ideals
  5. Ladder determinantal ideals
  6. Schubert determinantal ideals and poset ideals
  7. Chamfering ladders
  8. Mixed ladder determinantal ideals
  9. Knutson Schubert determinantal and poset ideals

Key Result

Theorem A

Let $\mathbb{k}$ be a perfect field of characteristic $p>0$ and $L$ be a ladder. For ${\bf t}=(t_1,\ldots,t_v) \in \mathbb{Z}_{>0}^v$ let $I_{\bf t}(L) \subseteq \mathbb{k}[L]$ be the corresponding mixed ladder determinantal ideal. The symbolic Rees algebra $\mathcal{R}^s\left(I_{{\bf t}}(L)\right)

Figures (2)

  • Figure 1: Ladder $L$
  • Figure 2: The subladders $L_1$, $L_2$, $L_3$, $L_4$ ordered from left to right

Theorems & Definitions (55)

  • Theorem A: \ref{['stronglyF']}
  • Theorem B: \ref{['ThmMixLadderSFP']}
  • Theorem C: \ref{['c:pif']}
  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Definition 3
  • ...and 45 more