On gromov-witten invariants of del pezzo surfaces

Mendy Shoval, Eugenii Shustin

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We compute Gromov-Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhäuser, Boston, 1995), pp. 81-148; L. Göttsche and R. Pandharipande, The quantum cohomology of blow-ups of P2 and enumerative geometry, J. Differential Geom. 48(1) (1998) 61-90], Gromov-Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math. 102 (2000) 53-84]. We solve the problem in two steps: (1) we consider curves on P 21, the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso-Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type P26,1, the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov-Witten invariants of P 27 via enumerative invariants of P2 6,1, using Vakil's extension of the Abramovich-Bertram formula.

Original languageEnglish
Article number1350054
JournalInternational Journal of Mathematics
Issue number7
StatePublished - Jun 2013


  • Enumerative geometry: Gromov-Witten invariants
  • deformations of curve singularities
  • moduli spaces of curves
  • rational surfaces


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