Logarithmic equivalence of Welschinger and Gromov-Witten invariants

By I. Itenberg, Viatcheslav Kharlamov, Eugenii Shustin. 2004

87 citations recorded by the source corpus.

The Welschinger numbers, a kind of a real analogue of the Gromov-Witten numbers that count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any generic collection of real points. Logarithmic equivalence of sequences is understood to mean the asymptotic equivalence of their logarithms. Such an equivalence is proved for the Welschinger and Gromov-Witten numbers of any toric Del Pezzo surface with its tautological real structure, in particular, of the projective plane, under the hypothesis that all, or almost all, the chosen points are real. A study is also made of the positivity of Welschinger numbers and their monotonicity with respect to the number of imaginary points.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0c8bdf60f0d494ef70614ebeeca49d72065632b6

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Citations in the corpus

Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions (2006, background): “3 where we develop the relationship with Welschinger invariants, for which many interesting calculations have already been carried out [8].”
Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3 (2011, methodology): “Remark 34 Theorem 7 covers all the cases studied in [11, 13, 14, 15, 20] (notice that the proof given in [15, Section 4.”
A Tropical Toolkit (2006, methodology): “Preprint submitted to Elsevier 3 April 2009 [33], the Sturmfels school [30], Itenberg, Kharlamov, and Shustin [ 18 ], Gathmann and Markwig [15], and Nishinou and Siebert [27] among many others.”
Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry (2003, methodology): “Remark 3.9. Using the results of this paragraph together with a modification of [10] to configurations admitting complex conjugated points due to E. Shustin (see [13]), I. Itenberg, V. Kharlamov and E. Shustin have been able to compute the polynomials χ 4 (T ) and χ 5 (T ), see [ 6 ].”