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
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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.
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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.
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3)
If h°(L)≥6 and L·L≥9, then s is an embedding.
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Andreatta, M., Ballico, E. On the adjunction process over a surface in char.p. Manuscripta Math 62, 227–244 (1988). https://doi.org/10.1007/BF01278981
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DOI: https://doi.org/10.1007/BF01278981