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
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 \(g\geq 11\).