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Covers of algebraic varieties. I: A general structure theorem, covers of degree 3, 4 and Enriques surfaces. (English) Zbl 0866.14009
The main result of this paper is a general structure theorem for Gorenstein coverings \(\rho:X\to Y\), namely finite flat maps between schemes (with \(Y\) integral) such that the scheme theoretic fibre \(\rho^{-1}(y)\) is Gorenstein for every \(y\) in \(Y\). More precisely, given a Gorenstein cover \(\rho:X\to Y\) of degree \(d\), denote by \(\mathcal E\) the cokernel of the natural inclusion \({\mathcal O}_Y\to\rho_*{\mathcal O}_X\): Then \(\mathcal E\) is a locally free rank \(d-1\) sheaf, \(X\) can be embedded in the \({\mathbb{P}}^{d-2}\)-bundle \({\mathbb{P}}(\mathcal E)\) in such a way that \({\mathcal O}_X(1)=\omega_{X|Y}\), and there exists a locally free resolution \[ 0\to N_{d-2}(-2-d)\to\ldots\to N_1(-2)\to {\mathcal O}_{\mathbb{P}}\to {\mathcal O}_X\to 0. \] Moreover, the ranks of the bundles \(N_i\) are computed and it is shown that the resolution is unique up to unique isomorphism. For \(d=3,4\), the result is made more explicit and sufficient conditions for the existence of smooth covers with given \(\mathcal E\) are given. These results are used to establish the existence of a coarse moduli space for Enriques surfaces with a polarization of degree \(4\) and to show that this moduli space is an irreducible unirational quasi-projective variety of dimension \(10\).
[For part II of this paper see G. Casnati, J. Algebr. Geom. 5, No. 3, 461-477 (1996)].
Reviewer: R.Pardini (Pisa)

14E20 Coverings in algebraic geometry
14J28 \(K3\) surfaces and Enriques surfaces