On higher-dimensional del Pezzo varieties

TL;DR

This paper explores higher-dimensional del Pezzo varieties, introducing an ADE classification, establishing bounds on their dimensions in type A, and classifying the maximal cases, thus extending the understanding of these algebraic varieties.

Contribution

It introduces an ADE classification for del Pezzo varieties, provides bounds on their dimensions in type A, and classifies the maximal del Pezzo varieties, advancing the theory of higher-dimensional algebraic varieties.

Findings

Dimension bound for non-conical del Pezzo varieties in type A

ADE classification of del Pezzo varieties

Complete classification of maximal del Pezzo varieties

Abstract

We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.