On the adjunction process over a surface in char.p

Abstract

Let Ks be the canonical bundle on a non singular projective surface S (over an algebraically closed field F, char F=p) and L be a very ample line bundle on S. Suppose (S,L) is not one of the following pairs: (P2,O(e)), e=1,2, a quadric, a scroll, a Del Pezzo surface, a conic bundle. Then 1) (Ks⊗L)2 is spanned at each point by global sections. Let\(\phi :S \to P^N _F \) be the map given by the sections Γ(Ks⊗L)2, and let φ=s o r its Stein factorization. 2) r:S→S′=r(S) is the contraction of a finite number of lines, Ei for i=1,...r, such that Ei·Ei=KS·Ei=−L·Ei=−1. 3) If h°(L)≥6 and L·L≥9, then s is an embedding.