Degenerations of complete collineations and geometric Tevelev degrees of ℙ𝑟

Abstract We consider the problem of enumerating maps 𝑓 of degree 𝑑 from a fixed general curve 𝐶 of genus 𝑔 to P r P^r satisfying incidence conditions of the form f ⁢ (p i) ∈ X i f (p₈) X₈, where p i ∈ C p₈ C are general points and X i ⊂ P r X₈^r are general linear spaces. We give a complete answer in the case where the X i X₈ are points, where the counts are known as the “Tevelev degrees” of P r P^r. These were previously known only when r = 1 r=1, when 𝑑 is large compared to r, g r, g, or virtually in Gromov–Witten theory. We also give a complete answer in the case r = 2 r=2 with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations.