Construction of algebraic covers

Abstract

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 ${\mathrm{Sym}}^{\bullet }\mathcal{F}$. 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 $\Phi :\mathcal{Q}\to \mathcal{R}(\mathcal{F})/⟨\mathcal{Q}⟩$ 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 ${\phi }_{\ast }{\mathcal{O}}_X$ admits an orthogonal decomposition. These are deformation of $S_3$-Galois branch covers.

References in corpus (1)

• Dihedral Galois covers of algebraic varieties and the simple cases