TY - JOUR

T1 - On gromov-witten invariants of del pezzo surfaces

AU - Shoval, Mendy

AU - Shustin, Eugenii

N1 - Funding Information:
The second author has enjoyed a support from the Israeli Science Foundation grant no. 448/09 and from the Hermann–Minkowski–Minerva Center for Geometry at Tel Aviv University. A substantial part of this work was done during the second author’s stay at the Mathematisches Forschungsinstitut Oberwolfach in the framework of the

PY - 2013/6

Y1 - 2013/6

N2 - 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.

AB - 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.

KW - Enumerative geometry: Gromov-Witten invariants

KW - deformations of curve singularities

KW - moduli spaces of curves

KW - rational surfaces

UR - http://www.scopus.com/inward/record.url?scp=84881017649&partnerID=8YFLogxK

U2 - 10.1142/S0129167X13500547

DO - 10.1142/S0129167X13500547

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AN - SCOPUS:84881017649

SN - 0129-167X

VL - 24

JO - International Journal of Mathematics

JF - International Journal of Mathematics

IS - 7

M1 - 1350054

ER -