Degenerations of complete collineations and geometric Tevelev degrees of P^r

We consider the problem of enumerating maps f of degree d from a fixed general curve C of genus g to Pr satisfying incidence conditions of the form f (pi) ∈ Xi, where pi ∈ C are general points and Xi ⊂ Pr are general linear spaces. We give a complete answer in the case where the Xi are points, where the counts are known as the "Tevelev degrees"of Pr . These were previously known only when r = 1, when d is large compared to r, g, or virtually in Gromov-Witten theory. We also give a complete answer in the case r = 2 with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations.