Refined count of real oriented rational curves
Thomas Blomme
TL;DR
The paper introduces a quantum index for oriented real curves in toric varieties, enabling a refined signed count of rational curves that generalizes previous results and connects to tropical invariants.
Contribution
It defines a quantum index for real curves, extending Mikhalkin's work to higher dimensions and linking refined enumerative invariants with tropical geometry.
Findings
Defined a quantum index related to amoeba area
Established a refined signed count of real rational curves
Connected refined invariants to tropical geometry
Abstract
We introduce a \textit{quantum index} for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen 2-form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the results from G. Mikhalkin arXiv:1505.04338 to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.
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Taxonomy
TopicsPolynomial and algebraic computation · Algebraic Geometry and Number Theory · Commutative Algebra and Its Applications