Quadruple covers of algebraic varieties

Let X and Y be varieties over a field k; pi : X --> Y is a quadruple cover of Y if pi(*)O(X) is a locally free, rank 4 O-Y-algebra. If char k not equa l 2, we see that pi(*)O(X) splits as O-Y + E where E is a locally free rank 3 sheaf over O-Y, which is locally the "trace zero" module. For each y is an element of Y, we therefore have a rank 4 associative, commutative algebr a over OY,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 E over a variety Y is determined by a totally decomposable section eta is an element of H-0 (boolean AND(2)S(2) E* X boolean AND(3) E). We then examine the case in which the section eta 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 E is split. In this context, Galois covers, which are covers induced by the acti on of a group of order 4 on the covering variety X, are also studied.