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Seshadri Regions and the Asymptotic Shape of Multigraded Regularity

Juliette Bruce, Lauren Cranton Heller, Mahrud Sayrafi, Alexandra Seceleanu

TL;DR

This work introduces the Seshadri region as a convex, multigraded invariant capturing all Seshadri constants simultaneously and uses it to describe the asymptotic behavior of multigraded Castelnuovo–Mumford regularity for powers of ideal sheaves on smooth projective toric varieties. The authors prove that reg_R(I^p)/p converges to the Seshadri region in the Painlevé–Kuratowski sense, with the limit being convex and, under Mori dream-space conditions, polyhedral. The methodology hinges on linking regularity to global generation, applying Fujita vanishing on blowups, and exploiting the projective-bundle analogy to extend to symmetric powers of vector bundles. The paper provides explicit polyhedral descriptions in toric settings and includes a detailed toric example illustrating the computational and conceptual framework.

Abstract

We introduce the Seshadri region of a subvariety, a convex region packaging the classical Seshadri constants with respect to every line bundle simultaneously. We develop the theory of Seshadri regions as a measure of positivity along subvarieties and apply it to determine asymptotic Castelnuovo-Mumford regularity for ideal powers and symmetric powers on smooth projective toric varieties.

Seshadri Regions and the Asymptotic Shape of Multigraded Regularity

TL;DR

This work introduces the Seshadri region as a convex, multigraded invariant capturing all Seshadri constants simultaneously and uses it to describe the asymptotic behavior of multigraded Castelnuovo–Mumford regularity for powers of ideal sheaves on smooth projective toric varieties. The authors prove that reg_R(I^p)/p converges to the Seshadri region in the Painlevé–Kuratowski sense, with the limit being convex and, under Mori dream-space conditions, polyhedral. The methodology hinges on linking regularity to global generation, applying Fujita vanishing on blowups, and exploiting the projective-bundle analogy to extend to symmetric powers of vector bundles. The paper provides explicit polyhedral descriptions in toric settings and includes a detailed toric example illustrating the computational and conceptual framework.

Abstract

We introduce the Seshadri region of a subvariety, a convex region packaging the classical Seshadri constants with respect to every line bundle simultaneously. We develop the theory of Seshadri regions as a measure of positivity along subvarieties and apply it to determine asymptotic Castelnuovo-Mumford regularity for ideal powers and symmetric powers on smooth projective toric varieties.

Paper Structure

This paper contains 16 sections, 31 theorems, 69 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background & Notation
  3. Algebraic Conventions
  4. Toric Varieties
  5. Seshadri Regions for Ideal Sheaves
  6. The Main Theorem & Convergence of Regions
  7. Upper Closed Limit, Seshadri Regions & Global Generation
  8. Global Generation & The Seshadri Region
  9. The Upper Closed Limit
  10. Lower Closed Limit, Seshadri Regions & Fujita Vanishing
  11. Fujita's Vanishing & The Seshadri Region
  12. The Lower Closed Limit
  13. Proofs of the main theorems
  14. A parallel theorem for vector bundles
  15. Example: blowing up the del Pezzo surface of degree 7
  16. ...and 1 more sections

Key Result

Lemma 3.2

Let $X$ be a complete algebraic variety. Let $\mathcal{I} \subseteq \mathcal{O}_X$ be an ideal sheaf and $\pi\colon\widetilde{X} \to X$ the blowup of $X$ along $\mathcal{I}$ with exceptional divisor $E$. Fix a subspace $V\subset N^1_\mathbb{R}\left(X\right)$.

Figures (7)

  • Figure 1: Left: The fan of $\mathcal{H}_2$. Right: $\mathop{\mathrm{Pic}}\nolimits(\mathcal{H}_2)$ with $\mathop{\mathrm{Nef}}\nolimits(\mathcal{H}_2)$ (dark blue).
  • Figure 2: The distance from the origin to the boundary of $\mathop{\mathrm{\mathbb{S}}}\nolimits(\mathcal{I})$ along a given ray is the reciprocal of the classical Seshadri constant $\varepsilon_{H}(\mathcal{I})$ for $H$ any ample class along the ray.
  • Figure 3: The fans of the Fano surfaces $F_{2,3}$ (left) and $F_{2,4}$ (right)
  • Figure 4: Seshadri region of $F_{2,3}$ at $\mathcal{I}$
  • Figure 5: The simplicial complex $\Delta$ used to compute the cohomology of $F_{2,3}$
  • ...and 2 more figures

Theorems & Definitions (73)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 63 more