Bounds on the torsion subgroups of Néron–Severi groups

Abstract:

Let $X \hookrightarrow \mathbb {P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. We give an explicit upper bound on the order of the torsion subgroup $(\operatorname {NS} X)_{\mathrm {tor}}$ of the Néron–Severi group of $X$. The bound is derived from an explicit upper bound on the number of irreducible components of the scheme $\operatorname {\mathbf {CDiv}}_n X$ parametrizing the effective Cartier divisors of degree $n$ on $X$. We also give an upper bound on the number of generators of $(\operatorname {NS} X)[\ell ^\infty ]$ uniform as $\ell \neq \mathrm {char} k$ varies.