Canonical models of surfaces of general type in positive characteristic (Q1121334)

The principal aim of the present paper is to prove an analogue of Bombieri's results [\textit{E. Bombieri}, Publ. Math., Inst. Hautes Étud. Sci. 42(1972), 171-219 (1973; Zbl 0259.14005)] in characteristic \(p>0.\) Let X be a minimal surface of general type over an algebraically closed field k of characteristic \(p>0.\) Theorem III:1.20 shows that then the linear systems \(| (m+1).K_ X|\) are base point free for \(m\geq 3\) or \(m=2\) and \(K^ 2\geq 2\) and moreover for \(m\geq 4\) or \(m=2\) and \(K^ 2\geq 2\) the divisor \((m+1)K_ X\) is very ample. The first basic step is to prove the following ``vanishing lemma'': Let X be as above \((char(k)=p>0)\). Then the first cohomology group of the sheaf \(-m.K_ X\) vanishes except possibly when \(m=1\), \(p=2\), \(\chi\) (\({\mathcal O}_ X)=1\) and X is (birationally) an inseparable double cover of a K3- surface or a rational surface. To prove the lemma, the author assumes non-vanishing and obtains various consequences to reach a contradiction. A failure of the non-vanishing gives rise to a construction of some special type of covering of degree \(p=char(k).\) In section I the author applies that construction and proves the auxiliary theorem I:2.4, which gives strong consequences from the assumption that there exists a numerically positive line bundle contained in the tangent bundle of X. A precise analysis of the surfaces which admit such a sort of bundle \({\mathcal L}\) with \(H^ 1(X,{\mathcal Q}_{{\mathcal L}^{-1}})\neq 0\) (theorem II:1.3) gives as a consequence the vanishing lemma (cf. theorem II:1.7). In section III the proof of theorem III:1.20 is completed.