Bounds on the Torsion Subgroups of Néron–Severi Groups

Abstract

Let 𝑋 ⤷ Pʳ be a smooth projective variety defined by homogeneous polynomials of degree ≤ 𝑑 over an algebraically closed field 𝑘. Let Pic 𝑋 be the Picard scheme of 𝑋, and let Pic⁰ 𝑋 be the identity component of Pic 𝑋. The Néron–Severi group scheme of 𝑋 is defined by NS 𝑋 = (Pic 𝑋)/(Pic⁰ 𝑋)ᵣₑ subscript d, and the Néron–Severi group of 𝑋 is defined by NS 𝑋 = (NS 𝑋)(𝑘). We give an explicit upper bound on the order of the finite group (NS 𝑋)ₜₒᵣ and the finite group scheme (NS 𝑋)ₜₒᵣ in terms of 𝑑 and 𝑟. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of 𝑋 and the finite group [mathematical equation]. We also show that (NS 𝑋)ₜₒᵣ is generated by (deg 𝑋 − 1)(deg 𝑋 − 2) elements in various situations.