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A Resolution of the Diagonal for Smooth Toric Varieties

Reginald Anderson

Abstract

Beilinson gave a resolution of the diagonal for complex projective spaces, which Bayer-Popescu-Sturmfels generalized to what they refer to as unimodular projective toric varieties. The unimodular condition in Bayer-Popescu-Sturmfels' convention is more restrictive than being smooth. In the smooth, non-unimodular setting, extra vertices appear in the finite cellular complex which Bayer-Popescu-Sturmfels previously used to resolve the diagonal for unimodular projective toric varieties. We use the floor function to assign monomial labelings in a convex manner, and show that this assignment is compatible with the graded algebra involving the irrelevant ideal to give a resolution of the diagonal for a smooth projective toric variety.

A Resolution of the Diagonal for Smooth Toric Varieties

Abstract

Beilinson gave a resolution of the diagonal for complex projective spaces, which Bayer-Popescu-Sturmfels generalized to what they refer to as unimodular projective toric varieties. The unimodular condition in Bayer-Popescu-Sturmfels' convention is more restrictive than being smooth. In the smooth, non-unimodular setting, extra vertices appear in the finite cellular complex which Bayer-Popescu-Sturmfels previously used to resolve the diagonal for unimodular projective toric varieties. We use the floor function to assign monomial labelings in a convex manner, and show that this assignment is compatible with the graded algebra involving the irrelevant ideal to give a resolution of the diagonal for a smooth projective toric variety.
Paper Structure (10 sections, 6 theorems, 77 equations, 12 figures, 2 tables)

This paper contains 10 sections, 6 theorems, 77 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Example of cellular resolution for $Bl_p \mathbb{P}^2$
  4. Resolving the diagonal for smooth, non-unimodular toric varieties in families
  5. Cokernel of $\partial^\epsilon_0$
  6. Example: Resolving the diagonal via deformation for $\mathrm{Bl}_p(\mathbb{P}^2)$ with the deformed complex $\mathcal{H}_L^\epsilon$
  7. Deforming the cellular complex $\mathcal{H}_L$
  8. Cokernel of $\partial^\epsilon_0$
  9. Moving around in the effective cone
  10. Smooth, non-unimodular example: Twice-Iterated Blow-up of $\mathbb{P}^2$ at a point

Key Result

Theorem 1.1

The complex $(\mathcal{F}^\bullet_{\mathcal{H}_L^\epsilon/L}, \partial^\epsilon)$ gives a resolution of $\mathcal{O}_\Delta$ as an object of $D^b_{Coh}(X_\Sigma \times X_\Sigma)$ for $X_\Sigma$ a smooth toric variety.

Figures (12)

  • Figure 1: $\mathbb{R} L \cap \mathcal{H}_L$ for $X_\Sigma$
  • Figure 2: Fundamental domain for $\faktor{(\mathbb{R} L \cap \mathcal{H}_L)}{L}$
  • Figure 3: The nef cone for $\mathrm{Bl}_p\mathbb{P}^2$
  • Figure 4: $\mathbb{R} L$, showing $\mathbb{R} L \cap \mathcal{H}_L \subset \mathbb{R}^4$
  • Figure 5: $\mathcal{H}_L^\epsilon$ for $\mathrm{Bl}_p \mathbb{P}^2$ deformed by $\mathcal{O}(D_1)+\mathcal{O}(D_4)$
  • ...and 7 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 9 more