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

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.