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Counting aCM Toric Bundles of Rank Two on the Veronese Surface

Yeonjae Hong, Sukmoon Huh

TL;DR

The paper addresses counting torus-equivariant rank-two aCM vector bundles on the Veronese surface by translating the problem to $d$-aCM bundles on $\mathbb{P}^2$ via the Veronese embedding. It leverages Klyachko's filtrations for toric bundles to convert the aCM condition into combinatorial data, leading to a complete enumeration formula $S(\mathbb{P}^2,d;2)=\frac{(d-1)d(d+1)(d+2)}{24}$ with explicit representatives. The approach provides a precise classification of non-split bundles up to twists and reveals a recurrence structure governing their growth, validated with small $d$ examples. This contributes a concrete finite-type counting result for a class of toric aCM bundles and demonstrates the power of equivariant filtration methods in toric cohomology calculations.

Abstract

We define the isomorphism classes of torus-equivariant rank 2 arithmetically Cohen-Macaulay (aCM) vector bundles on the Veronese surface, up to a twist by the hyperplane class, and count them. Our approach makes use of Klyachko's description of toric vector bundles via filtrations and the associated cohomology computation. We also describe several representative bundles.

Counting aCM Toric Bundles of Rank Two on the Veronese Surface

TL;DR

The paper addresses counting torus-equivariant rank-two aCM vector bundles on the Veronese surface by translating the problem to -aCM bundles on via the Veronese embedding. It leverages Klyachko's filtrations for toric bundles to convert the aCM condition into combinatorial data, leading to a complete enumeration formula with explicit representatives. The approach provides a precise classification of non-split bundles up to twists and reveals a recurrence structure governing their growth, validated with small examples. This contributes a concrete finite-type counting result for a class of toric aCM bundles and demonstrates the power of equivariant filtration methods in toric cohomology calculations.

Abstract

We define the isomorphism classes of torus-equivariant rank 2 arithmetically Cohen-Macaulay (aCM) vector bundles on the Veronese surface, up to a twist by the hyperplane class, and count them. Our approach makes use of Klyachko's description of toric vector bundles via filtrations and the associated cohomology computation. We also describe several representative bundles.

Paper Structure

This paper contains 5 sections, 7 theorems, 52 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Toric Geometry and Equivariant Sheaves
  3. Basics on the toric variety
  4. Equivariant torsion free sheaves
  5. d-aCM bundles over P2

Key Result

Theorem 1.1

The number $\mathrm{S}(\mathbb{P}^2,d;2)$ of non-split $d$-aCM vector bundles of rank $2$ with equivariant structure on $\mathbb{P}^2$, up to twist by $\mathcal{O}_{\mathbb{P}^2}(dt)$ with $t\in \mathbb{Z}$, is given by

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Example 2.5
  • Corollary 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • ...and 14 more