## Algebra

•  François Greer, IAS
•  A tale of two Severi curves
•  02/17/2021
•  4:00 PM - 5:00 PM
•  Online (virtual meeting) (Virtual Meeting Link)
•  Laure Flapan (flapanla@msu.edu)

Let $(S,L)$ be a general polarized K3 surface with $c_1(L)^2=2g-2$. A general member of the linear system $|L|\simeq \mathbb P^g$ is a smooth curve of genus $g$. For $0\leq h\leq g$, define the Severi variety $V_h(S,L)\subset |L|$ to be the locus of curves with geometric genus $\leq h$. As expected, $V_h(S,L)$ has dimension $h$. We consider the case $h=1$, where the Severi variety is a (singular) curve. Our first result is that the geometric genus of $V_1(S,L)$ goes to infinity with $g$; we give a lower bound $\sim e^{c\sqrt{g}}$. Next we consider the analogous question for Severi curves of a rational elliptic surface, and give a polynomial upper bound instead. Modular forms play a central role in both arguments. Passcode: MSUALG

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