Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-30T02:04:28.225Z Has data issue: false hasContentIssue false
  • English
  • Français

COMBINED COUNT OF REAL RATIONAL CURVES OF CANONICAL DEGREE 2 ON REAL DEL PEZZO SURFACES WITH $K^2=1$

Part of: Surfaces and higher-dimensional varieties Projective and enumerative geometry

Published online by Cambridge University Press:  14 June 2022

S. Finashin  and
V. Kharlamov Open the ORCID record for V. Kharlamov [Opens in a new window]
S. Finashin
Affiliation:
Department of Mathematics, Middle East Technical University, Dumlupinar Bul. 1, Universiteler Mah, 06800 Ankara, Turkey
V. Kharlamov*
Affiliation:
Université de Strasbourg et IRMA (CNRS), 7 rue René-Descartes, 67084 Strasbourg Cedex, France
*
kharlam@math.unistra.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$, but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.

Keywords

Real del Pezzo surfacesBertini involutionPin-structuresEnumerative invariantsSigned countWelschinger weight

MSC classification

Primary: 14N10: Enumerative problems (combinatorial problems)
Secondary: 14P25: Topology of real algebraic varieties 14J26: Rational and ruled surfaces 14N15: Classical problems, Schubert calculus
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 123 - 148
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atiyah, M., On analytic surfaces with doublepoints . Proc. Roy. Soc. London. Ser. A 247 (1958),237–244.Google Scholar
Brugallé, E., Floor diagrams relative to a conic, and GW-Winvariants of del Pezzo surfaces . Adv. Math., 279 ( 2015 ), 438–500.CrossRefGoogle Scholar
Brugallé, E., Surgery of real symplectic fourfolds andWelschinger invariants . Journal of Singularities, 17 ( 2018 ), 267–294.Google Scholar
Brugallé, E., On the invariance of Welschingerinvariants . Algebra i Analiz, 32 ( 2020 ),1–20.Google Scholar
Brugallé, E., Puignau, N. On Welschinger invariants of symplectic 4-manifolds . Comment. Math. Helv. 90 ( 2015 ), 905–938.CrossRefGoogle Scholar
Conway, J.H., Sloane, N.J.A., Sphere Packings, Lattices and Groups. Springer,1988.CrossRefGoogle Scholar
Finashin, S., Kharlamov, V., Two kinds of real lines on real del Pezzo surfaces of degree1 . Selecta Math. (N.S.) 27 (2021), no. 5, Article No. 83.Google Scholar
Göttsche, L. and Pandharipande, R., The quantum cohomology of blow-ups of ${\mathbb{P}}^2$ and enumerative geometry. J. Diff.Geom. 48 (1998), no. 1, 61–90.Google Scholar
Itenberg, I., Shustin, E.,Kharlamov, V., Welschingerinvariants of real del Pezzo surfaces of degree $\ge 2$ . International J.Math. 26 (2015), no. 8, 1550060.CrossRefGoogle Scholar
Shevchishin, V.V., On the local Severiproblem . Int. Math. Res. Not. 2004, no. 5, 211–237.CrossRefGoogle Scholar
Shevchishin, V.V., Kharlamov, V. Anti-symplectic involutions on rational symplectic 4-manifolds. Contemporary Math. 772 (2021), 165–172.Google Scholar
Shoval, M. and Shustin, E., Counting curves of any genus on ${\mathbb{P}}_7^2$ .Google Scholar
Vakil, R., Counting curves on rationalsurfaces . Manuscripta Math. 102 (2000),53–84.CrossRefGoogle Scholar
Welschinger, J.-Y., Invariants of real symplectic 4-manifolds andlower bounds in real enumerative geometry . Invent. Math. 162 (2005), no.1, 195–234.CrossRefGoogle Scholar