38406501359372282063949 and All That: Monodromy of Fano Problems

A Fano problem is an enumerative problem of counting r-dimensional linear subspaces on a complete intersection in Pn 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 Fr(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(E6) or W(Dk).