TY - JOUR

T1 - Welschinger invariants of real del Pezzo surfaces of degree ≥ 2

AU - Itenberg, Ilia

AU - Kharlamov, Viatcheslav

AU - Shustin, Eugenii

N1 - Publisher Copyright:
© 2015 World Scientific Publishing Company.

PY - 2015/7/28

Y1 - 2015/7/28

N2 - We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D Pic(X), through any generic collection of - DKX - 1 real points lying on a connected component of the real part ℝX of X one can trace a real rational curve C |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov-Witten invariants. Our approach consists in a conversion of Shoval-Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich-Bertram formula for Gromov-Witten invariants into formulas computing real enumerative invariants.

AB - We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D Pic(X), through any generic collection of - DKX - 1 real points lying on a connected component of the real part ℝX of X one can trace a real rational curve C |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov-Witten invariants. Our approach consists in a conversion of Shoval-Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich-Bertram formula for Gromov-Witten invariants into formulas computing real enumerative invariants.

KW - Abramovich-Bertram-Vakil formula

KW - Caporaso-Harris formula

KW - Real rational curves

KW - Welschinger invariants

KW - enumerative geometry

UR - http://www.scopus.com/inward/record.url?scp=84938210471&partnerID=8YFLogxK

U2 - 10.1142/S0129167X15500603

DO - 10.1142/S0129167X15500603

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AN - SCOPUS:84938210471

SN - 0129-167X

VL - 26

JO - International Journal of Mathematics

JF - International Journal of Mathematics

IS - 8

M1 - 1550060

ER -