Ballico, E.; Chiantini, L.; Monti, V. On the adjunction mapping for surfaces of Kodaira dimension \(\leq{}0\) in char. p. (English) Zbl 0762.14004 Manuscr. Math. 73, No. 3, 313-318 (1991). 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) Cited in 3 Documents 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 Keywords:adjunction; algebraic surface; spannedness; very ampleness; adjoint bundle Citations:Zbl 0663.14010; Zbl 0702.14010 PDFBibTeX XMLCite \textit{E. Ballico} et al., Manuscr. Math. 73, No. 3, 313--318 (1991; Zbl 0762.14004) Full Text: DOI EuDML References: [1] Andreatta, M.–Ballico, E.:On the adjunction process over a surface in char. p, Manuscripta Math.62, 227–244 (1988) · Zbl 0698.14034 [2] Andreatta, M.–Ballico, E.:Classification of projective surfaces with small sectional genus: char p, Rend. Sem. Mat. Univ. Padova, (to appear) · Zbl 0748.14013 [3] Andreatta, M.–Ballico, E.:On the adjunction process over a surface in char. p: the singular case, J. reine angew. Math. (to appear) · Zbl 0721.14027 [4] Beltrametti, M.–Francia, P.–Sommese, A. J.:On Reider’s method and higher order embeddings, Duke Math. J.,58, 425–439 (1989) · Zbl 0702.14010 [5] Beltrametti, M.–Sommese, A. J.:Zero cycles and k-th order embeddings of smooth projective surfaces, Cortona Proceedings on Projective Surfaces and their Classification, Symposia Mathematica, INDAM, Academic Press (to appear) · Zbl 0827.14029 [6] Bombieri, E.:Canonical models of surfaces of general type, Publ. Math. I.H.E.S.42, 447–495 (1973). [7] Bombieri, E.:Methods of algebraic geometry in char. p and their applications, in: Algebraic Surfaces C.I.M.E., Cortona 1977, p. 57–96, Liguori (Napoli), 1981 [8] Bombieri, E.–Mumford, D.:Enriques’ classification of surfaces in char. p, II, in: Complex Analysis and Algebraic Geometry, p. 23–42, Cambridge University Press, 1977 · Zbl 0348.14021 [9] Ekedahl, T.:Canonical models of surfaces of general type in positive characteristic, Publ. I.H.E.S.67, 97–144 (1988) · Zbl 0674.14028 [10] Forster, O.–Hirschowitz, A.–Schneider, M.:Type de scindage généralisé pour les fibrés stables, in: Vector Bundles and Differential Equations, p. 65–81, Progress in Math. vol. 7, Birkhauser, 1979 [11] Katsura, T.–Oort, F.:Families of supersingular abelian surfaces, preprint. · Zbl 0636.14017 [12] Lazarsfeld, R.:Brill-Noether-Petri without degenerations, J. Diff. Geom.23, 299–307 (1986) · Zbl 0608.14026 [13] Oort, F.:Which abelian surfaces are products of elliptic curves, Math. Ann.214, 35–47 (1975) · Zbl 0291.14014 [14] Reider, I.:Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math.127, 309–316 (1988) · Zbl 0663.14010 [15] Sommese, A. J.:Hyperplane sections of projective surfaces, I. The adjunction mapping, Duke Math. J.46, 377–401 (1979) · Zbl 0415.14019 [16] A. Van de Ven:On the 2-connectedness of very ample divisors on a surface, Duke Math. J.46, 403–407 (1979) · Zbl 0458.14003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.