Symmetric power of higher dimensional varieties

TL;DR

This paper investigates the geometric and combinatorial properties of symmetric powers of smooth varieties, including their Picard groups, divisor classes, and stratifications based on singularities and partitions.

Contribution

It provides a detailed description of the Picard and divisor class groups of symmetric powers and introduces a combinatorial stratification framework based on partitions.

Findings

Describes Picard and divisor class groups of symmetric powers

Provides a stratification of symmetric powers by singular loci

Connects stratification to combinatorial data of partitions

Abstract

We study several properties of the symmetric power $S^mX$ of a smooth variety $X$. We describe the Picard and divisor class groups of $S^mX$ when $X$ is projective. We give a complete description of the stratification of $S^mX$ by iterated singular locus in terms of some combinatorial data regarding partitions of the integer $m.$ This gives a new viewpoint of a natural stratification of $S^mX$ by multiplicities.