Moduli Spaces for Rings and Ideals

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The investigator studies moduli spaces of finite, flat covers and line bundles of those covers. Important basic examples include orders in number fields (finite flat covers of the integers) and finite covers of the complex projective line. The methods involve working over an arbitrary base scheme, so for example one gets results about both the number theoretic and geometric examples above. The project is to find moduli spaces that have explicit descriptions and reasonable geometry so that they can be worked with concretely. Since the work of Gauss in 1801, polynomials have been use to study number systems that are bigger than the usual counting numbers 1,2,3.., For example, a larger number system might also include the square root of 2, which cannot be found among 1,2,3... When we include the square root of 2, it is a quadratic extension of the usual numbers, and if we had included the cube root of 2 it would have been a cubic extension of the usual numbers. This work tries to understand what the possible low degree extensions of the usual numbers are by working explicitly with polynomials that are related to the extensions. This then allows one to make computations regarding the larger number systems by reducing them to easier computations about polynomials.