## Student Algebraic Geometry Seminar: Geometry of the moduli space of curves of genus $g$

Seminar | April 17 | 4-5 p.m. | 891 Evans Hall

Ritvik Ramkumar, UC Berkeley

Department of Mathematics

Given a variety $X$ it's a basic question to ask if it's rational i.e. admits a birational map $P^n \to X$ for some $n$. For curves and surfaces there are explicit criterion that determine when a variety is rational. Answering this question in higher dimensions is much more difficult. In light of this, it's easier to ask for a weaker question: Does there exist a dominant rational map $P^n\to X$. In this case, $X$ is said to be unirational. Over the complex numbers, every unirational curve or surface is rational. For curves this is a consequence of the Riemann-Hurwitz Formula.

A very important geometric object is $M_g$, the moduli space of curves of genus $g$. We can ask for what $g$, if any, is $M_g$ rational or unirational. In 1915, Severi proved that $M_g$ is unirational for $g\leq 10$ and conjectured that it's unirational for all $g$. However, in 1987, it was shown that $M_g$ is of general type if $g\geq 24$. In this talk I will review classical examples of unirational varieties, outline a modern proof of Severi's result and describe related conjectures and results. I will finish by describing how the geometry of $M_g$ changes when $g\geq 11$.

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