Seminormal toric varieties

TL;DR

This work addresses describing seminormal toric varieties in a purely combinatorial framework, relaxing normality. It introduces semisaturated monoids and a fan-based construction with attached groups to encode seminormality and good torus actions. The authors prove an equivalence of categories between the category of seminormal toric varieties with good action, $\mathfrak{TorVar}^{sn}$, and the category of fans with attached groups, $\mathfrak{Fan}^{gr}$. This extends the classical normal toric correspondence and provides a streamlined set of gluing data for a broader class of toric varieties.

Abstract

In this paper, we provide a combinatorial description of seminormal toric varieties. The corresponding combinatorial object is a fan equipped with a collection of groups assigned to each cone. This framework introduces a more general class of toric varieties than classical normal toric varieties, while having simpler combinatorial data compared to general non-normal toric varieties.

Figures (6)

• Figure 1: Illustration of \ref{['ExempleNotSemisaturated']}
• Figure 2: Illustration of \ref{['ExempleSemisaturated']}
• Figure 3: Illustration of \ref{['EternelExempleAlpha']}
• Figure 4: Illustration of \ref{['ExempleDualiteConeVarNormal']}
• Figure 5: A fan with attached groups

Theorems & Definitions (54)

• Theorem
• Lemma 1.0.1: REID2001703 Theorem 3.2 b)
• Definition 1.0.2
• Example 1.0.3
• Definition 1.0.4
• Remark 1.0.5
• Lemma 1.0.6
• proof
• Lemma 1.0.7: Swan1992Gubeladze, Lemma 6.5
• proof