Bounds on the torsion subgroup schemes of Néron–Severi group schemes

Abstract

Let X↪Pr be a smooth projective variety defined by homogeneous polynomials of degree ≤d over an algebraically closed field. Let Pic X be the Picard scheme of X. Let Pic0X be the identity component of Pic X. The Néron–Severi group scheme of X is defined by NSX=(PicX)/(Pic0X)red. We give an explicit upper bound on the order of the finite group scheme (NSX)tor in terms of d and r. As a corollary, we give an upper bound on the order of the finite group πe´t1(X,x0)torab. We also show that the torsion subgroup (NSX)tor of the Néron–Severi group of X is generated by at most (deg⁡X−1)(deg⁡X−2) elements in various situations.