Welschinger invariants of small non-toric Del Pezzo surfaces

作者：Itenberg Ilia^*; Kharlamov Viatcheslav; Shustin Eugenii

来源：Journal of the European Mathematical Society, 2013, 15(2): 539-594.

DOI：10.4171/JEMS/367

• Tropical curves
• real rational curves
• enumerative geometry
• Welschinger invariants
• Caporaso-Harris formula

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 %26lt;= 1 pairs of conjugate imaginary points, where q + 2s %26lt;= 5, and the real quadric blown up at s %26lt;= 1 pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil%26apos;s recursive formula [ 22] for Gromov-Witten invariants of these surfaces and generalizes our recursive formula [ 12] for purely real Welschinger 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.

• 出版日期2013