Mathematics faculty and graduate students are invited to attend the PhD Defense of Matthew Williams.

Title:  Pseudostable Hodge Classes

Advisor:  Dr. Renzo Cavalieri
Committee:  Dr. Chris Peterson, Dr. Maria Gillespie, Dr. Dustin Tucker

Abstract:   Intersection theory is a branch of algebraic geometry that studies the intersections of geometric objects, and intersection theory on the moduli spaces of stable curves has long been researched. Pseudostable curves differ from their stable counterparts in that cusp singularities are allowed, but elliptic tails are disallowed, and this leads to a different intersection theory. Hodge classes are fundamental objects relating to the genera of curves, and until recently, intersections involving pseudostable Hodge classes have been relatively unexplored.

We study the relationship between Hodge classes on the moduli spaces of pseudostable and stable curves given by the contraction morphism T, which maps elliptic tails to cusps. A useful identity of stable Hodge classes is Mumford’s relation, and while it does not hold in the pseudostable case, we show that one can express a similar relation using the pullback via T solely in terms of strata and psi classes. Further, we organize the combinatorial structure of the pullback of products of two pseudostable Hodge classes and obtain an explicit comparison of arbitrary pseudostable and stable quadratic Hodge integrals.