Welschinger invariants of small non-toric Del Pezzo surfaces

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)539-594
Number of pages56
JournalJournal of the European Mathematical Society
Issue number2
StatePublished - 2013


  • Caporaso-Harris formula
  • Enumerative geometry
  • Real rational curves
  • Tropical curves
  • Welschinger invariants

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics


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