1998/03/09 by Aaron Bertram, Bertram, Aaron, Michael Thaddeus +1 · 3 citations
Mathematics · #14H99 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.AG #math.DG #math.SG #msc:14H99
paper · pdf · doi:10.48550/arxiv.math/9803026
28 pages; LaTeX2e, with packages amsfonts, latexsym, eepic
openalex publication_date 1998/03/09 · arxiv created 2001/01/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [3/4 g, g-1). Otherwise, they still give a formula modulo third order terms. Explicit generators and relations are also given unless d is in [4/5 g - 3/5, g-1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically a formula of Harris-Tu for the Chern numbers of determinantal varieties. The case d = g-1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with Givental's work is also explained.