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 $\le d$ over an algebraically closed field. Let $\mathbf{P}\mathbf{i}\mathbf{c}X$ be the Picard scheme of $X$. Let $$\mathbf{P}\mathbf{i}\mathbf{c}$^0 X$betheidentitycomponentof$$\mathbf{P}\mathbf{i}\mathbf{c}$, X$. The N'eron–Severi group scheme of $X$ is defined by $$\mathbf{N}\mathbf{S}$, X = ($\mathbf{P}\mathbf{i}\mathbf{c}$, X)/($\mathbf{P}\mathbf{i}\mathbf{c}$^0 X)_{$\mathrm{r}\mathrm{e}\mathrm{d}$}$. We give an explicit upper bound on the order of the finite group scheme $(\mathbf{N}\mathbf{S}X{)}_{\mathrm{t}\mathrm{o}\mathrm{r}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the finite group ${\pi }_{\mathrm{e}\mathrm{t}}^1(X,x_0{)}_{\mathrm{t}\mathrm{o}\mathrm{r}}^{\mathrm{a}\mathrm{b}}$. We also show that the torsion subgroup $(\mathrm{N}\mathrm{S}X{)}_{\mathrm{t}\mathrm{o}\mathrm{r}}$ of the N'eron–Severi group of $X$ is generated by less than or equal to $(\mathrm{d}\mathrm{e}\mathrm{g}X-1)(\mathrm{d}\mathrm{e}\mathrm{g}X-2)$ elements in various situations.