  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.
