Double Ramification Cycles and Tautological Classes

Algebraic geometry is a branch of mathematics that studies geometric properties of spaces defined by polynomial equations. One of the key objects in algebraic geometry is the moduli space of curves, a geometric space whose points correspond to types of algebraic curves. This moduli space has many connections to other areas of mathematics and to physics (where the algebraic curves are the strings appearing in string theory). One way to study the moduli space of curves is via its intersection theory, the study of how certain loci in the moduli space (those corresponding to curves with specific properties) intersect each other. In this project, the PI will investigate various problems relating to a fundamental intersection-theoretic class on the moduli space of curves: the double ramification cycle. This project will also provide research opportunities for graduate students, who will be trained in the methods of the field. The PI will study two main groups of problems dealing with the intersection theory of the moduli space of curves. First, the PI will develop improved formulas for the logarithmic double ramification cycle, a refinement of the double ramification cycle constructed and studied in the last several years by Holmes, Ranganathan, Schwarz, and others. The PI and his coauthors recently developed an approach to computing this cycle, and the primary goal of this project is to make this approach more effective so that it can be applied to localization computations in logarithmic geometry. Second, the PI will investigate assorted connections that have surfaced in recent years between the double ramification cycle and other tautological classes. This includes developing a theory of log tautological relations, defining and computing an orbifold version of the double ramification cycle, and attempting to prove a one-dimensional socle result for the tautological ring of the moduli space of bridgeless curves. 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.