## 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)

### MSC:

 14E20 Coverings in algebraic geometry 14J28 $$K3$$ surfaces and Enriques surfaces