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

2023/07/31 by Carl Lian, Lian, Carl
Mathematics · #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2308.00046

openalex publication_date 2023/07/31 · openalex created_date 2023/08/03 · openalex updated_date 2026/07/28

Abstract

We consider the problem of enumerating maps f of degree d from a fixed general curve C of genus g to ℙ^r satisfying incidence conditions of the form f(p_i)∈ X_i, where p_i∈ C are general points and X_i⊂ℙ^r are general linear spaces. We give a complete answer in the case where the X_i are points, where the counts, the ``Tevelev degrees'' of ℙ^r, 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.

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