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On the adjunction mapping for surfaces of Kodaira dimension \(\leq{}0\) in char. p. (English) Zbl 0762.14004

Let \(S\) be a smooth complete connected algebraic surface \(S\) of Kodaira dimension \(\kappa(S)\leq 0\) defined over an algebraically closed field of any characteristic and let \(L\) be an ample line bundle on \(S\). The authors prove the spannedness, the very ampleness, and more generally the \(k\)-spannedness, of the adjoint bundle \(K_ S\otimes L\) under some conditions on \((S,L)\) depending on \(\kappa(S)\) and \(k\). These conditions are similar to the ones required in the complex case by Reider’s method [I. Reider, Ann. Math., II. Ser. 127, No. 2, 309-316 (1988; Zbl 0663.14010)] and his generalization due to M. Beltrametti, P. Francia and A. J. Sommese [Duke Math. J. 58, No. 2, 425-439 (1989; Zbl 0702.14010)].
Reviewer: A.Lanteri (Milano)

MSC:

14C20 Divisors, linear systems, invertible sheaves
14G15 Finite ground fields in algebraic geometry
14J25 Special surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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References:

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