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  • David Hahn

My current area of research is in quadruple covers of algebraic varieties, i.e. flat, finite maps between varieties which are generically four to one. More specifically, we seek data on a variety Y with which we may build a quadruple cover. This is in the spirit of double covers, which are determined by a divisor L on Y and a global section D of O_X(-2L). We seek a map on the rings of functions from O_Y to O_X. Locally, this is the question of determining rank four associative algebras over a given field. We have found that such algebras are parameterised by an affine cone over a G(2,6), the grassmanian of two dimensional subspaces of a six dimensional space, with vertex the algebra given by the fat point k[x,y,z]/(x,y,z)². We then describe the maps without reference to a local basis, thus allowing our analysis to sheafify.

In addition to the general case, we have examined Galois covers as well. These are covers which are induced by the action of a group of order four on X. We have also looked at the parameterising variety of rank 3 Lie algebras. The equations for this variety arise in a similar manner as those for the local rings of a quadruple cover. However, instead of relations arising from associativity conditions, they come from imposing the Jacobi identity.

I am currently continuing my work on quadruple covers, trying to find the equations for the ramification and brach loci in terms of the building data. I have also begun a study of grobner bases in an attempt to develop research areas which would be appropriate for undergraduates.