An enumerative application of the quantum cohomology of a stack
-- Charles Cadman, January 30, 2004

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Kontsevich's recursion for the number of rational degree d plane curves
passing through 3d-1 general points is an application of the theory of
stable maps and quantum cohomology to enumerative geometry.  I will
discuss an extension of this theory to stacks -- due to Abramovich, Graber,
and Vistoli -- and apply it to the enumerative geometry of curves with
tangency conditions.

This application requires the following stack construction.  Given a
scheme Y and an effective Cartier divisor D, there is a stack
over Y with the property that a morphism f: C --> Y from a smooth curve
into Y lifts to the stack precisely when f*(D) is a multiple of 2
(assuming C intersects D properly).  In the case where D is a smooth
cubic in the projective plane, the genus 0 Gromov-Witten invariants of
this stack are enumerative and can be computed from the WDVV equations.