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A
tropical computation of refined toric invariants
Thomas Blomme |
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Geometry & Topology 29 (2025) 3129–3185
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Abstract |
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In 2015, G Mikhalkin introduced a refined count for real rational curves in toric surfaces. The counted curves have to pass through some real and complex points located on the toric boundary of the surface, and the count is refined according to the value of a so-called quantum index. This count happens only to depend on the number of complex points on each toric divisor, leading to an invariant. First, we give a way to compute the quantum index of any oriented real rational curve, getting rid of the previously needed “purely imaginary” assumption on the complex points. Then we use the tropical geometry approach to relate these classical refined invariants to tropical refined invariants, defined using Block–Göttsche multiplicity. This generalizes the result of Mikhalkin relating both invariants in the case where all the points are real, and the result of the author where complex points are located on a single toric divisor. |
Keywords
refined invariants, tropical geometry, enumerative geometry
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Mathematical Subject Classification
Primary: 14H99, 14M25, 14N10, 14T90
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References |
Publication
Received: 25 March 2021
Revised: 16 July 2023
Accepted: 1 October 2023
Published: 22 September 2025
Proposed: Lothar Göttsche
Seconded: Dan Abramovich, Mark Gross
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Authors |
| Thomas Blomme | ||
| Institut de Mathématiques de Jussieu
– Paris Rive Gauche Sorbonne Université Paris France |
Département de mathématiques et
applications Ecole Normale Supérieure Paris France |
Institut de Mathématiques Université de Neuchâtel Neuchâtel Switzerland |
| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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