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.
Categories: math.AG