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On elliptic surfaces in characteristic p. (English) Zbl 0553.14019

In this paper we study the question to what extend the theory of elliptic surfaces over the complex number field \({\mathbb{C}}\) can be extended or has a good analogy in positive characteristics. The main difficulty comes from the existence of wild fibres in positive characteristics. - In the present paper we first prove the following theorem: If S is an algebraic elliptic surface defined over an algebraically closed field k of characteristic p \(\geq 0\) with Kodaira dimension one, then the m-ple canonical mapping \(\Phi_{| mK_ S|}\) gives the unique structure of the elliptic surface for every \(m\geq 14\). - We also give an example that the number 14 is the best possible, if \(char(k)\neq 2,3.\) S. Iitaka showed that a similar theorem holds for analytic elliptic surfaces and obtained the estimate \(m\geq 86\) [J. Math. Soc. Japan 22, 247-261 (1970; Zbl 0188.534)]. Our theorem is new even for the case \(k={\mathbb{C}}\). To prove the theorem, we need to study an elliptic surface \(f: S\to P^ 1\) with \(\chi (S,0_ S)=0.\) We show that multiple fibres of such an elliptic surface satisfy certain numerical conditions (theorem 3.3). - Next we study wild fibres. We obtain the following theorem: Let mD be a wild fibre of an elliptic surface \(f:S\to C\) where D is an ordinary elliptic curve or of type \(I_ n\) (that is, a cycle of rational curves). Then there exists an element \(\alpha \in H^ 1(S,0_ S)\), a covering \(\pi_ 1:S^{(1)}\to S\) and an elliptic surface \(f_ 1: S^{(1)}\to C^{(1)}\) such that the corresponding multiple fibre of \(f_ 1\) has a form \(m^{(1)}D^{(1)}\) with \(m^{(1)}p=m.\) By a finite succession of this process the wild fibre is reduced to a tame fibre. - If D is a supersingular elliptic curve, we also have a similar result. In this case we need to take a flat cover (§6, case III). In §7 we give the process to reduce a tame fibre mD to a non multiple fibre, if D is an elliptic curve or of type \(I_ n\). In §8 several examples of wild fibres are given.
In the present paper we do not consider a multiple fibre mD where D is of type \({\mathbb{G}}_ a\) which appear only in characteristic \(p>0\). Recently we obtained a similar result in this case, if \(p\geq 5\). This will be discussed in a forthcoming paper. - In §9 and §10 we consider deformation and lifting of elliptic surfaces and algebraic surfaces. In §9 we show that Kodaira dimension of an algebraic surface is invariant under deformation and lifting, though plurigenera may be not invariant. In §10 we show that the genus of the base curve of an elliptic surface is invariant under deformation and lifting.

MSC:

14J25 Special surfaces
14G20 Local ground fields in algebraic geometry
14D15 Formal methods and deformations in algebraic geometry
14J10 Families, moduli, classification: algebraic theory

Citations:

Zbl 0188.534
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References:

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