Computing Picard Schemes

TL;DR

This paper introduces an algorithm to explicitly compute the torsion part of the Picard scheme of a smooth projective variety, along with applications to fundamental groups and cohomology, advancing computational algebraic geometry.

Contribution

It provides the first explicit algorithms for computing the torsion component of the Picard scheme and related invariants, including the fundamental group abelianization and Galois module structures.

Findings

Explicit description of $ ext{Pic}^ au X$ as a closed subscheme

Algorithms for computing the abelianization of the étale fundamental group

Determination of Galois module structures of étale cohomology groups

Abstract

We present an algorithm to compute the torsion component $\mathrm{Pic}^\tau X$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^\tau X$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on $\mathrm{Pic}^\tau X$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric \'etale fundamental group $\pi^{\mathrm{{e}t}}_1(X_{\bar{k}}, x)^{\mathrm{ab}}$. Moreover, we determine the Galois module structure of the first \'etale cohomology groups $H^1_{\mathrm{{e}t}}(X_{\bar{k}}, \mathbb{Z}/n\mathbb{Z})$ without requiring $n$ to be prime to the characteristic of $k$.