Bounds on the Torsion Subgroups of N\'eron-Severi Group Schemes

TL;DR

This paper establishes explicit bounds on the torsion subgroup of the Néron-Severi group scheme for smooth projective varieties, linking geometric properties with algebraic invariants, and providing bounds on related étale fundamental groups.

Contribution

It provides the first explicit bounds on the torsion of the Néron-Severi group scheme in terms of the degree and dimension of the variety, and relates these bounds to étale fundamental groups.

Findings

Bound on the order of the torsion subgroup of the Néron-Severi group scheme.

Upper bound on the order of the torsion subgroup of the étale fundamental group.

The torsion subgroup of the Néron-Severi group is generated by at most (deg X -1)(deg X - 2) elements.

Abstract

Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$. Let $\mathbf{Pic}^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron--Severi group scheme of $X$ is defined by $\mathbf{NS}\, X = (\mathbf{Pic}\, X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$. We give an explicit upper bound on the order of the finite group scheme $(\mathbf{NS}\, X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the finite group $\pi^1_{\mathrm{et}}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that the torsion subgroup $(\mathrm{NS}\, X)_{\mathrm{tor}}$ of the N\'eron--Severi group of $X$ is generated by less than or equal to $(\mathrm{deg}\, X -1)(\mathrm{deg}\, X - 2)$ elements in various situations.