Constructions and Applications of Compactified Moduli

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Algebraic geometry is the field of mathematics that studies objects defined by polynomial equations, called varieties. One of the aims of moduli theory is to try to understand properties of a given variety by thinking about the space of all possible shapes such a variety can have. This space of all shapes is the moduli space of a variety of a given numerical type. For example, the cubic threefold is a variety in 4-space given by one equation of degree 3. Much modern progress in understanding moduli is due to degeneration techniques – the idea of understanding a given variety by thinking how it can break up into simpler objects. This project aims to further the understanding of the properties of some of the moduli spaces central to algebraic geometry: the moduli of cubic threefolds, the moduli of complex curves (that is, objects that from close up look like complex numbers), and other related moduli spaces. In particular, the PI will study how such varieties can degenerate, or break up, and will aim to understand the behavior of geometric constructions under such degenerations. The project will support the work of the PI and his PhD students on these topics. The project aims to better understand the geometry of compactifications of some of the moduli spaces ubiquitous in algebraic geometry: moduli of of curves, of curves with a differential, and of cubic threefolds. The PI will also explore applications to problems in Teichmueller dynamics and in classical algebraic geometry. The PI will apply the compactification of the strata of differentials (that is, of moduli of curves with a differential with prescribed multiplicities of zeroes and poles) that he constructed with other researchers. The PI will also investigate the birational geometry of strata and will approach the central orbit classification questions in Teichmueller dynamics by applying degeneration techniques of algebraic geometry. The PI will apply real-normalized differentials, the geometry of which he studied, and the degeneration techniques he co-developed to work towards a sharp upper bound for the number of cusps of plane curves. Likewise, the PI will co-investigate the relations among cones of divisors and log-MMP for various compactifications of the moduli space of cubic threefolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.