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

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

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
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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

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 123 - 148
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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