Quadruple covers of algebraic varieties

Let X and Y be varieties over a field k; π : X → Y is a quadruple cover of Y if π*script O signX is a locally free, rank 4 script O signY-algebra. If char k ≠ 2, we see that π*script O signX splits as script O signY ⊕ε where ε is a locally free rank 3 sheaf over script O signY, which is locally the "trace zero" module. For each y ∈ Y, we therefore have a rank 4 associative, commutative algebra over script O signY,y. We find that these algebras are parametrized by an affine cone over the Grassmanian G(2, 6) with vertex corresponding to the algebra k[x, y, z]/(x, y, z)2. We then show that a quadruple cover with trace zero module ε over a variety Y is determined by a totally decomposable section η ∈ H0(∧2S2 E* ⊗ ∧3 E). We then examine the case in which the section η has no zeros. Here, each rank 4 algebra may be associated to a pencil of conics. As a special case of this, we look at the work of G. Casnati and T. Ekedahl on Gorenstein covers, and we show that their analysis is the subcase where the pencil of conics has length 4 base locus. Finally, we study the case in which the trace zero module ε is split. In this context, Galois covers, which are covers induced by the action of a group of order 4 on the covering variety X, are also studied.