Computing Picard Schemes

We present an algorithm to compute the torsion component ${\mathrm{P}\mathrm{i}\mathrm{c}}^{\tau }X$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe ${\mathrm{P}\mathrm{i}\mathrm{c}}^{\tau }X$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on ${\mathrm{P}\mathrm{i}\mathrm{c}}^{\tau }X$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group ${\pi }_1^{\mathrm{e}\mathrm{t}}(X_{\bar{k}},x{)}^{\mathrm{a}\mathrm{b}}$. Moreover, we determine the Galois module structure of the first étale cohomology groups $H_{\mathrm{e}\mathrm{t}}^1(X_{\bar{k}},\mathbb{Z}\mathrm{/}n\mathbb{Z})$ without requiring $n$ to be prime to the characteristic of $k$.