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.
Dedicated to Gert-Martin Greuel in occasion of his 70th birthday
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhäuser, Basel (1986)
Degtyarev, A., Kharlamov, V.: Topological properties of real algebraic varieties: Rokhlin’s way. Russ. Math. Surv. 55 (4), 735–814 (2000)
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)
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)
Graber, T., Kock, J., Pandharipande, R.: Descendant invariants and characteristic numbers. Am. J. Math. 124 (3), 611–647 (2002)
Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer, Berlin (2007)
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)
Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariants of real del Pezzo surfaces of degree ≥ 3. Math. Ann. 355 (3), 849–878 (2013)
Itenberg, I., Kharlamov, V., Shustin, E.: Relative enumerative invariants of real nodal del Pezzo surfaces. Preprint at arXiv:1611.02938 (2016)
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/S0129167X15500603
Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariant revisited. Preprint at arXiv:1409.3966 (2014)
Shustin, E.: On manifolds of singular algebraic curves. Sel. Math. Sov. 10 (1), 27–37 (1991)
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)]
Shustin, E.: On higher genus Welschinger invariants of Del Pezzo surfaces. Int. Math. Res. Not. 2015, 6907–6940 (2015). doi:10.1093/imrn/rnu148
Vakil, R.: Counting curves on rational surfaces. Manuscripta Math. 102 (1), 53–84 (2000)
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)
Welschinger, J.-Y.: Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. Invent. Math. 162 (1), 195–234 (2005)
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)
Welschinger, J.-Y.: Towards relative invariants of real symplectic four-manifolds. Geom. Asp. Funct. Anal. 16 (5), 1157–1182 (2006)
Welschinger, J.-Y.: Enumerative invariants of strongly semipositive real symplectic six-manifolds. Preprint at arXiv:math.AG/0509121 (2005)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Shustin, E. (2017). On Welschinger Invariants of Descendant Type. In: Decker, W., Pfister, G., Schulze, M. (eds) Singularities and Computer Algebra. Springer, Cham. https://doi.org/10.1007/978-3-319-28829-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-28829-1_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28828-4
Online ISBN: 978-3-319-28829-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)