38406501359372282063949 & all that: Monodromy of Fano Problems

A Fano problem is an enumerative problem of counting $r$-dimensional linear subspaces on a complete intersection in ${\mathbb{P}}^n$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes $F_r(X)$ as the complete intersection $X$ varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups $W(E_6)$ or $W(D_k)$.