An imaginary refined count for some real rational curves

In 2015, Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through a set consisting of real points and pairs of complex conjugated points chosen generically on the toric boundary of the surface. He then proved that the result of this refined count depend...

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Bibliographic Details
Published in:Beiträge zur Algebra und Geometrie 2023-09, Vol.64 (3), p.721-751
Main Author: Blomme, Thomas
Format: Article
Language:English
Subjects:
Algebra
Algebraic Geometry
Convex and Discrete Geometry
Geometry
Invariants
Mathematics
Mathematics and Statistics
Original Paper
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Summary:In 2015, Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through a set consisting of real points and pairs of complex conjugated points chosen generically on the toric boundary of the surface. He then proved that the result of this refined count depends only on the number of pairs of complex conjugated points on each toric divisor. Using the tropical geometry approach and the correspondence theorem, we address the computation of the refined count when the pairs of complex conjugated points are chosen purely imaginary and belonging to the same component of the toric boundary. Despite the non-genericity, we relate this refined count for purely imaginary values to the refined invariant of Mikhalkin for generic values. That allows us to extend the relation between these classical refined invariants and the tropical refined invariants from Block–Göttsche.
ISSN:0138-4821
2191-0383
DOI:10.1007/s13366-022-00653-z