Moduli spaces and maps between them

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The PI proposes to study the geometry of moduli of curves, of abelian varieties, of Prym varieties, and of cubic threefolds. He will study the questions of injectivity of maps between these spaces (the Torelli problem), and of describing the images of such maps (the Schottky problem). The PI will further develop the technique of meromorphic differentials with real periods that he developed with Krichever to study the geometry of the moduli space of curves, and singularities of plane curves. Using his results with Hulek on the locus of intermediate Jacobians of cubic threefolds, the PI will attempt to define an extended tautological ring for compactifications of the moduli space of abelian varieties, and to study the classes of natural loci in it, as a possible inductive approach to degenerations of abelian varieties. The PI will also aim to obtain an explicit solution to the classical Schottky problem in genus 5, by using motivation and his results on string scattering amplitudes. Further, the PI will attempt to use his characterization, with Krichever, of Prym varieties to approach the Prym-Torelli problem.In algebraic geometry, one basic question is to describe the set of all objects of a given type. Given an algebraic variety (a zero set of a system of polynomial equations), one can try to deform it, by deforming the defining equations, and ask what is the space of deformations, or ask what is the space of varieties that can be deformed to a given one. These parameter spaces for varieties are called moduli spaces, and turn out to often have a rich geometric structure themselves. Moreover, in many instances there are constructions associating to a variety of one kind a variety of a different kind (for example the Jacobian of a Riemann surface), and these constructions define maps of one moduli space to another. It is natural to ask whether these maps preserve all the information (that is, whether the image determines the source - whether the map is injective; this is known as the Torelli problem) and whether all varieties can be obtained by such a construction (that is, whether the map is surjective; if not, describing the image is the Schottky problem). The proposed project aims to provide a better understanding and more explicit description of the structure of these geometric moduli spaces and relations among them. The PI proposes to work on some longstanding open questions in moduli theory, and will also work on developing new tools and techniques for studying moduli spaces.