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Measuring rationality of Schwede--Takagi pairs

Pat Lank, Peter McDonald, Sridhar Venkatesh

TL;DR

The paper provides a derived-category framework to characterize rational singularities of Schwede–Takagi pairs, establishing that the level of $\omega_Y$ with respect to the multiplier submodule $\mathcal{J}(\omega_Y, \mathcal{I}^c)$ equals $1$ precisely when the pair is rational (after passing to completions). In the affine locally complete intersection setting, this level is always finite and serves as a computable invariant that measures the pair’s failure to be rational, thereby generalizing and recovering results of Lank–Venkatesh (2024) and connecting singularity theory with derived-category generation. The approach yields a numerical, homological diagnostic for singularities of pairs and has practical computability via Macaulay2 in suitable cases, offering a new tool for exploring the interaction between the ambient geometry and the defining ideal in the minimal model program context.

Abstract

Our work begins by providing a derived characterization of rational singularities for pairs in the sense of Schwede--Takagi. This characterization both generalizes and independently obtains a result of Lank--Venkatesh in the special case of varieties with rational singularities. As an application, we introduce a categorical invariant that measures the failure of rationality for pairs on affine varieties which are locally complete intersections.

Measuring rationality of Schwede--Takagi pairs

TL;DR

The paper provides a derived-category framework to characterize rational singularities of Schwede–Takagi pairs, establishing that the level of with respect to the multiplier submodule equals precisely when the pair is rational (after passing to completions). In the affine locally complete intersection setting, this level is always finite and serves as a computable invariant that measures the pair’s failure to be rational, thereby generalizing and recovering results of Lank–Venkatesh (2024) and connecting singularity theory with derived-category generation. The approach yields a numerical, homological diagnostic for singularities of pairs and has practical computability via Macaulay2 in suitable cases, offering a new tool for exploring the interaction between the ambient geometry and the defining ideal in the minimal model program context.

Abstract

Our work begins by providing a derived characterization of rational singularities for pairs in the sense of Schwede--Takagi. This characterization both generalizes and independently obtains a result of Lank--Venkatesh in the special case of varieties with rational singularities. As an application, we introduce a categorical invariant that measures the failure of rationality for pairs on affine varieties which are locally complete intersections.
Paper Structure (6 sections, 4 theorems, 9 equations)

This paper contains 6 sections, 4 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generation
  4. Dualizing complexes
  5. Schwede--Takagi rationality
  6. Results

Key Result

Proposition 3.1

$(Y,\mathcal{I}^c)$ has rational singularities if, and only if, $\mathcal{O}_Y \in \langle \mathbf{R} f_\ast \mathcal{O}_{\widetilde{Y}}(\lfloor c \cdot G \rfloor) \rangle_1$ for some log resolution $f\colon \widetilde{Y}\to Y$ of $(Y,\mathcal{I}^c)$.

Theorems & Definitions (9)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Example 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof