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Some geometric applications of a differential equation in characteristic $$p>0$$ to the theory of algebraic surfaces. (with the cooperation og J. Sturnfield and J. Lang). (English) Zbl 0561.14018
Contemp. Math. 13, 335-347 (1982).
This paper is concerned with the Zariski surfaces (i.e. smooth projective surfaces over an algebraic closed field k of characteristic p$$>0$$ endowed with a purely inseparable rational dominant map of degree $$p: X\to {\mathbb{P}}^ 2).$$ When X is a generic Zariski surface (i.e. only mild singularities are involved in the explicit construction of X) formulae are given for genus, étale Betti numbers, Artin invariant. The last is obtained from the solutions of a differential equation over k that in many case can be solved effectively. A condition for two generic Zariski surfaces to be isomorphic is given. This leads to the construction of moduli spaces. Finally the notion of Zariski surface is generalised by introducing the new concept of $$\beta$$-equivalence (the surfaces can be joined by a dominant purely inseparable map). - This paper is not a complete exposition, it outlines the theory of Zariski surfaces and gives only sketch of proofs. On the other hand it gives some examples and announces several open problems and conjectures.
For the entire collection see [Zbl 0496.00006].
Reviewer: G.Christol

##### MSC:
 14J25 Special surfaces 14G15 Finite ground fields in algebraic geometry 12H25 $$p$$-adic differential equations 14J10 Families, moduli, classification: algebraic theory
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