38406501359372282063949 & all that: Monodromy of Fano problems

Abstract: A Fano problem is an enumerative problem of counting linear subspaces on complete intersections. Some familiar examples are finding the number of lines on a cubic surface, and finding the number of lines on the intersection of $2$ quadrics in $\mathbb{P}^4$. Suppose a general complete intersection of type $[d]=(d_1, ..., d_s)$ in $\mathbb{P}^n$ contains finitely many $r$-planes. To this Fano problem, described by the triple $([d],n,r)$, one can associate a group $G_{[d],n,r}$, the monodromy group of the Fano problem; it describes the permutations of $r$-planes on a complete intersection of type $[d]$, as the complete intersection varies. I will show that $G_{[d],n,r}$ is either a symmetric or an alternating group for almost all Fano problems with an explicit list of exceptions, and describe the monodromy groups of the exceptional problems. An interesting feature of this computation is that it avoids any local calculations, which seems necessary to get the result in full generality. This is joint work with Sachi Hashimoto.

algebraic geometry

Audience: researchers in the topic

Comments: Discussion during the talk will be at tinyurl.com/2020-05-01-a (and this will be deleted in 3 days).

Stanford algebraic geometry seminar

Series comments: The seminar was online for a significant period of time, but for now is solely in person. More seminar information (including slides and videos, when available): agstanford.com