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 $\le d$. We give explicit upper bounds on the order of the torsion subgroup $(\mathrm{NS}X{)}_{\mathrm{tor}}$ of the Néron-Severi group of $X$. The bounds are derived from an explicit upper bound on the number of irreducible components of either the Hilbert scheme ${\mathbf{Hilb}}_{Q}X$ or the scheme ${\mathbf{CDiv}}_{n}X$parametrizing the effective Cartier divisors of degree $n$ on $X$. We also give an upper bound on the number of generators of $(\mathrm{NS}X)[{\ell }^{\infty }]$ uniform as $\ell \ne \mathrm{char}k$ varies.

17 pages