Bounds on the Torsion Subgroups of Néron-Severi Group Schemes

Abstract

Let $X\hookrightarrow {\mathbb{P}}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\le d$ 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 Néron--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 ${\pi }_{\mathrm{et}}^1(X,x_0{)}_{\mathrm{tor}}^{\mathrm{ab}}$. We also show that the torsion subgroup $(\mathrm{NS}X{)}_{\mathrm{tor}}$ of the Néron--Severi group of $X$ is generated by less than or equal to $(\mathrm{deg}X-1)(\mathrm{deg}X-2)$ elements in various situations.