The quantum cohomology of blow-ups of P^2 and enumerative geometry

Abstract

We compute the Gromov-Witten invariants of the projective plane blown up in r general points. These are determined by associativity from r+1 intial values. Applications are given to the enumeration of rational plane curves with prescribed multiplicities at fixed general points. We show that the numbers are enumerative if at least one of the prescribed multiplicities is 1 or 2. In particular, all the invariants for r<=8 (the Del Pezzo case) are enumerative.

AMS-LaTeX, 26 pages

References in corpus (3)

• The quantum cohomology of homogeneous varieties
• Counting rational curves with multiple points and Gromov-Witten invariants of blow-ups
• Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

Cited by in corpus (5)

• Logarithmic equivalence of Welschinger and Gromov-Witten invariants
• Welschinger invariants of real del Pezzo surfaces of degree $\ge 2$
• Rational Curves on the Space of Determinantal Nets of Conics
• On the asymptotic enumerativity property for Fano manifolds
• Mixed anomalies of chiral algebras compactified to smooth quasi-projective surfaces