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Blowup algebras of determinantal ideals in prime characteristic

Alessandro De Stefani, Jonathan Montaño, Luis Núñez-Betancourt

Abstract

We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results we develop the notion of $F$-split filtrations and symbolic $F$-split ideals.

Blowup algebras of determinantal ideals in prime characteristic

Abstract

We study when blowup algebras are -split or strongly -regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results we develop the notion of -split filtrations and symbolic -split ideals.

Paper Structure

This paper contains 23 sections, 64 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Graded algebras
  4. Methods in prime characteristic
  5. Local cohomology and Castelnuovo-Mumford regularity
  6. Filtrations and blowup algebras
  7. Generators of defining equations of $F$-split blowup algebras
  8. $F$-split filtrations
  9. $F$-split filtrations of ideals
  10. $F$-split blowup algebras
  11. Depth and regularity of $F$-split filtrations
  12. Symbolic $F$-split ideals and symbolic powers
  13. Symbolic and ordinary powers of determinantal ideals
  14. Ideals of minors of a generic matrix
  15. Ideals of minors of a symmetric matrix
  16. ...and 8 more sections

Key Result

Theorem 1

Let $K$ be an $F$-finite field of prime characteristic $p>0$. Let $X$ be a generic matrix, $Y$ be a generic symmetric matrix, $Z$ be a generic skew-symmetric matrix, and $W$ be a generic Hankel matrix. For an integer $t>0$ we have

Theorems & Definitions (159)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • ...and 149 more