Let $Y$ be an algebraic variety, $\mathcal{ F}$ a locally free sheaf of $\mathcal{ O}_Y$-modules, and $\mathcal{ R}(\mathcal{ F})$ the $\mathcal{ O}_Y$-algebra $\operatorname{ Sym}^\bullet \mathcal{ F}$. In this paper we study local properties of sheaves of $\mathcal{ O}_{ \mathcal{ R}(\mathcal{ F})}$-ideals $\mathcal{ I}$ such that $\mathcal{ R}(\mathcal{ F}))/\mathcal{ I}$ is an algebraic cover of $Y$. Following the work of Miranda for triple covers, for $\mathcal{ Q}$ a direct summand of $\mathcal{ R}(\mathcal{ F})$, we say that a morphism $\Phi\colon \mathcal{ Q}\rightarrow\mathcal{ R}(\mathcal{ F})/\langle\mathcal{ Q}\rangle$ is a covering homomorphism if it induces such an ideal. As an application we study in detail the case of Gorenstein covering maps of degree $6$ for which the direct image of $\varphi_*\mathcal{ O}_X$ admits an orthogonal decomposition. These are deformation of $S_3$-Galois branch covers.
Categories: math.AG math.AC