Computing Picard Schemes
Hyuk Jun Kweon, Madhavan Venkatesh
TL;DR
This work delivers an explicit, computable framework to extract the torsion part of the Picard scheme, $\mathrm{Pic}^\tau X$, as a closed subscheme of projective space endowed with a group-scheme structure. It develops a systematic Grassmannian-based pipeline—via Stiefel/Plücker coordinates and Gotzmann bounds—to realize divisors $mH$ and their numerical equivalence classes, then forms the quotient to obtain $\mathrm{Pic}_{mH} X \cong \mathrm{Pic}^\tau X$ along with explicit addition, inverse, and identity maps. The method yields concrete algorithms for computing the abelianization of the geometric étale fundamental group and the $\mathrm{Aut}(\bar{k}/k)$-module structure of $H^1_{\mathrm{et}}(X_{\bar{k}}, \mathbb{Z}/n\mathbb{Z})$, including non-prime-to-characteristic cases and $p$-power phenomena that influence non-reduced Picard structures. Beyond the Picard-torsion, the paper also details how to compute Albanese varieties and torsion in NS, tying the Picard-scheme data to key homological invariants with explicit, coordinate-based formulas. This establishes a robust computational paradigm for higher-dimensional moduli spaces and their cohomological avatars.
Abstract
We present an algorithm to compute the torsion component $\mathrm{Pic}^τX$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^τX$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on $\mathrm{Pic}^τX$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group $π^{\mathrm{{e}t}}_1(X_{\bar{k}}, x)^{\mathrm{ab}}$. Moreover, we determine the Galois module structure of the first étale 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 $k$.