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On Welschinger Invariants of Descendant Type

  • Eugenii ShustinEmail author
Chapter
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Abstract

We introduce enumerative invariants of real del Pezzo surfaces that count real rational curves belonging to a given divisor class, passing through a generic conjugation-invariant configuration of points and satisfying preassigned tangency conditions to given smooth arcs centered at the fixed points. The counted curves are equipped with Welschinger-type signs. We prove that such a count does not depend neither on the choice of the point-arc configuration nor on the variation of the ambient real surface. These invariants can be regarded as a real counterpart of (complex) descendant invariants.

Keywords

del Pezzo surfaces Descendant invariants Real enumerative geometry Real rational curves Welschinger invariants 

Subject Classification:

Primary 14N35; Secondary 14H10 14J26 14P05 

Notes

Acknowledgements

The author has been supported by the grant no. 1174-197.6/2011 from the German-Israeli Foundations, by the grant no. 176/15 from the Israeli Science Foundation and by a grant from the Hermann Minkowski–Minerva Center for Geometry at the Tel Aviv University.

References

  1. 1.
    Abramovich, D., Bertram, A.: The formula 12 = 10 + 2 × 1 and its generalizations: counting rational curves on F 2. In: Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000). Contemporary Mathematics, vol. 276, pp. 83–88. American Mathematical Society, Providence, RI (2001)Google Scholar
  2. 2.
    Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhäuser, Basel (1986)CrossRefzbMATHGoogle Scholar
  3. 3.
    Degtyarev, A., Kharlamov, V.: Topological properties of real algebraic varieties: Rokhlin’s way. Russ. Math. Surv. 55 (4), 735–814 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Georgieva, P., Zinger, A.: Enumeration of real curves in \(\mathbb{C}P^{2n-1}\) and a WDVV relation for real Gromov–Witten invariants. Preprint at arXiv:1309.4079 (2013)Google Scholar
  5. 5.
    Georgieva, P., Zinger, A.: A recursion for counts of real curves in \(\mathbb{C}CP^{2n-1}\): another proof. Preprint at arXiv:1401.1750 (2014)Google Scholar
  6. 6.
    Graber, T., Kock, J., Pandharipande, R.: Descendant invariants and characteristic numbers. Am. J. Math. 124 (3), 611–647 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer, Berlin (2007)zbMATHGoogle Scholar
  8. 8.
    Gudkov, D.A., Shustin, E.I.: On the intersection of the close algebraic curves. In: Topology (Leningrad, 1982). Lecture Notes in Mathematics, vol. 1060, pp. 278–289. Springer, Berlin (1984)Google Scholar
  9. 9.
    Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariants of real del Pezzo surfaces of degree ≥ 3. Math. Ann. 355 (3), 849–878 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Itenberg, I., Kharlamov, V., Shustin, E.: Relative enumerative invariants of real nodal del Pezzo surfaces. Preprint at arXiv:1611.02938 (2016)Google Scholar
  11. 11.
    Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariants of real del Pezzo surfaces of degree ≥ 2. Int. J. Math. 26 (6) (2015). doi:10.1142/S0129167X15500603Google Scholar
  12. 12.
    Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariant revisited. Preprint at arXiv:1409.3966 (2014)Google Scholar
  13. 13.
    Shustin, E.: On manifolds of singular algebraic curves. Sel. Math. Sov. 10 (1), 27–37 (1991)zbMATHGoogle Scholar
  14. 14.
    Shustin, E.: A tropical approach to enumerative geometry. Algebra i Analiz 17 (2), 170–214 (2005) [English Translation: St. Petersburg Math. J. 17, 343–375 (2006)]Google Scholar
  15. 15.
    Shustin, E.: On higher genus Welschinger invariants of Del Pezzo surfaces. Int. Math. Res. Not. 2015, 6907–6940 (2015). doi:10.1093/imrn/rnu148MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Vakil, R.: Counting curves on rational surfaces. Manuscripta Math. 102 (1), 53–84 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Welschinger, J.-Y.: Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry. C. R. Acad. Sci. Paris, Sér. I 336, 341–344 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Welschinger, J.-Y.: Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. Invent. Math. 162 (1), 195–234 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Welschinger, J.-Y.: Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants. Duke Math. J. 127 (1), 89–121 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Welschinger, J.-Y.: Towards relative invariants of real symplectic four-manifolds. Geom. Asp. Funct. Anal. 16 (5), 1157–1182 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Welschinger, J.-Y.: Enumerative invariants of strongly semipositive real symplectic six-manifolds. Preprint at arXiv:math.AG/0509121 (2005)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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