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The quantum cohomology of blow-ups of ${\bf P}\sp 2$ and enumerative geometry

1998; Lehigh University; Volume: 48; Issue: 1 Linguagem: Inglês

10.4310/jdg/1214460607

1945-743X

Lothar Göttsche, Rahul Pandharipande,

Algebraic structures and combinatorial models

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.