Computing Picard Schemes

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 $k$.