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Zbl 0674.14028
Ekedahl, Torsten
Canonical models of surfaces of general type in positive characteristic. (English)
[J] Publ. Math., Inst. Hautes Étud. Sci. 67, 97-144 (1988). ISSN 0073-8301; ISSN 1618-1913

The principal aim of the present paper is to prove an analogue of Bombieri's results [{\it 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 $\vert (m+1).K\sb X\vert$ are base point free for $m\ge 3$ or $m=2$ and $K\sp 2\ge 2$ and moreover for $m\ge 4$ or $m=2$ and $K\sp 2\ge 2$ the divisor $(m+1)K\sb X$ is very ample. \par 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 $-m.K\sb X$ vanishes except possibly when $m=1$, $p=2$, $\chi$ (${\cal O}\sb X)=1$ 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. \par 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 ${\cal L}$ with $H\sp 1(X,{\cal Q}\sb{{\cal L}\sp{-1}})\ne 0$ (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.
[A.Iliev]

MSC 2000:: *14J29 Surfaces of general type
14C20 Divisors, linear systems, invertible sheaves

Keywords: base point free linear systems; minimal surface of general type; characteristic p; divisor; vanishing lemma

Citations: Zbl 0259.14005

Cited in: Zbl 0871.14017 Zbl 0808.14036 Zbl 0807.14006 Zbl 0769.14006

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