2020/10/15 by Adam Afandi, Afandi, Adam
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2010.07521
openalex publication_date 2020/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using Atiyah-Bott localization on the space of stable maps to the stack quotient [ℙ1/ℤ2], we find recursions that determine all Hodge integrals with descendent insertions at one marked point on the hyperelliptic locus Hg, 2g + 2 ⊆ Mg, 2g + 2. The initial conditions required for our recursions are gravitational descendents at one marked point, which are known to be (1)/(2). We discover a new structure concerning these intersection numbers: for a fixed monomial of λ-classes, the resulting family of hyperelliptic Hodge integrals is polynomial in g. We formulate a conjecture concerning the log-concavity of the coefficients of these polynomials. Lastly, we turn our recursions into a non-linear system of partial differential equations for the generating functions of hyperelliptic Hodge integrals.