A remark on the Grothendieck-Lefschetz theorem about the Picard group

Abstract

Let K be an algebraically closed field of arbitrary characteristic. The term “variety” always means here an irreducible algebraic variety over K. The notations and the terminology are borrowed in general from EGA [4].

Skew-symmetric matrices and Palatini scrolls

“…Now recall that, for m ≥ 5, the group Pic(Y ) is generated by H Y by the theorem Grothendieck-Lefschetz, while, for m = 4, the cokernel of the natural restriction map Pic(P(U )) → Pic(Y ) is torsion-free (we refer for instance to [2]). It follows that Hence the linear system |M | is a g s k on the curve Y , with s ≥ 2.…”

“…We consider the dual Euler sequence on P(V ), twisted by K φ (2), and take global sections. We obtain an inclusion: (1)).…”

“…For instance, in 1891, Castelnuovo [11] considered the case m = 3 and n = 4, namely the case of nets of linear complexes in P 4 . The locus of lines which are centers of the linear complexes belonging to a general net of linear complexes in P 4 , or, in modern language, the degeneracy locus of a general morphism φ : O 3 P 4 → P 4 (2), is the projected Veronese surface in P 4 .…”

“…Given a general map φ : O m P(V ) → P(V ) (2), we consider φ as the point [φ] of the Grassmann variety G(m, ∧ 2 V * ) parametrizing m-dimensional vector subspaces of ∧ 2 V * . Since a linear change of coordinate on U does not change the degeneracy locus X φ , we have a rational map:…”

On ample divisors

“…for every zn > 0, and then apply the theorem of [2] (in a slightly modified form) to deduce that a is injective and Coker (a) is torsion-free. Since Indeed, the coherence of F {m) comes from [7], expose VIII, Corollary VIΠ-Π-3.…”