Sara Torelli:

Abstract: Compact Riemann surfaces are two-dimensional real manifolds with a complex structure. In 1857, Riemann was studying how to vary a complex structure on a compact Riemann surface, and he first suggested that such variations should be called “moduli.” A word that algebraic geometers use now habitually and like to study from many perspectives.  

Given the beautiful correspondence between compact Riemann surfaces and smooth complex complete algebraic curves,
we talk in the modern language of “moduli of curves.” In this seminar, I will introduce them as complex varieties, and I will convince you that studying holomorphic forms, namely sections of the cotangent bundle and its exterior powers, is a powerful tool for understanding some of their geometric features. To do this, I will survey the results on moduli of stable curves, which is an established and deeply studied perspective, and suggest a new direction of research on moduli of smooth curves, explaining my first result in this direction. The result states that holomorphic one forms do not appear on the moduli space of smooth projective curves.

Karl Christ:

Abstract: Severi varieties parametrize integral curves of fixed geometric genus in a given linear system on a surface. In this talk, I will discuss the classical question of whether Severi varieties are irreducible and its relation to the irreducibility of other moduli spaces of curves. I will try to indicate how new, tropical methods can be used to answer such irreducibility questions. The new results are from ongoing joint work with Xiang He and Ilya Tyomkin.