TY - JOUR

T1 - Welschinger invariants of small non-toric Del Pezzo surfaces

AU - Itenberg, Ilia

AU - Kharlamov, Viatcheslav

AU - Shustin, Eugenii

PY - 2013

Y1 - 2013

N2 - We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at q real and s 1 pairs of conjugate imaginary points, where q C 2s 5, and the real quadric blown up at s 1 pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil's recursive formula [22] for Gromov-Witten invariants of these surfaces and generalizes our recursive formula [12] for purely realWelschinger invariants of real toric Del Pezzo surfaces. As a consequence, we prove the positivity of the Welschinger invariants under consideration and their logarithmic asymptotic equivalence to genus zero Gromov-Witten invariants.

AB - We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at q real and s 1 pairs of conjugate imaginary points, where q C 2s 5, and the real quadric blown up at s 1 pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil's recursive formula [22] for Gromov-Witten invariants of these surfaces and generalizes our recursive formula [12] for purely realWelschinger invariants of real toric Del Pezzo surfaces. As a consequence, we prove the positivity of the Welschinger invariants under consideration and their logarithmic asymptotic equivalence to genus zero Gromov-Witten invariants.

KW - Caporaso-Harris formula

KW - Enumerative geometry

KW - Real rational curves

KW - Tropical curves

KW - Welschinger invariants

UR - http://www.scopus.com/inward/record.url?scp=84873813476&partnerID=8YFLogxK

U2 - 10.4171/JEMS/367

DO - 10.4171/JEMS/367

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AN - SCOPUS:84873813476

SN - 1435-9855

VL - 15

SP - 539

EP - 594

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 2

ER -