Refined count for rational tropical curves in arbitrary dimension

Blomme, Thomas

doi:10.1007/s00208-021-02287-3

Cited by 4 publications

(4 citation statements)

References 19 publications

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“…In the case of toric surfaces, I. Itenberg and G. Mikhalkin proved in [14] that the count of tropical curves with fixed genus and degree passing through the right number of points with refined multiplicity is invariant. These results were generalized in various settings, see for instance [21], [1], [10], [3], [4].…”

Section: Refined Invariants For Curves In Linear Systemsmentioning

confidence: 84%

Tropical curves in abelian surfaces II: enumeration of curves in linear systems

Blomme¹

2022

Preprint

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In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus g curves in a fixed linear system in an abelian surface to a tropical count. To do this, we relate the linear system defined by a complex curve to certain integrals of 1-forms over cycles in the curve. We then give an expression for the tropical multiplicity provided by the correspondence theorem, and prove the invariance for the associated refined multiplicity, thus introducing refined invariants of Block-Göttsche type in abelian surfaces.

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Section: Refined Invariants For Curves In Linear Systemsmentioning

confidence: 84%

Tropical curves in abelian surfaces II: enumeration of curves in linear systems

Blomme¹

2022

Preprint

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“…Real refined enumeration. In some situations, namely in [Mik17] and [Blo21], the tropical refined invariants are connected to some refined real enumerative invariants. More precisely, we perform a signed count of real rational curves with prescribed tangency conditions on the toric boundary of the considered variety, refined by the value of a suitable quantum index.…”

Section: Descendant Invariantsmentioning

confidence: 99%

“…In this way, their computations yield the refined invariants versions of Severi degrees considered in [BG16b] and [IM13] rather than the usual ones. The meaning of these tropical refined invariants remains an intense area of studies [GS14, Bou19, Mik17, Blo21] that have already been generalized to different settings [Blo22, GS19, BS19, SS18], but remain quite mysterious.…”

Section: Introductionmentioning

confidence: 99%

Floor diagrams and enumerative invariants of line bundles over an elliptic curve

Blomme¹

2023

Compositio Math.

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We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are $\mathbb {C} P^1$ -bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block–Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

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“…Such refined invariants have since been extended to many other situations. See for instance [10], [22], [1], [21], [5]. The setting of floor diagrams has also been adapted to enable the computation of some of these refined invariants in [2].…”

Section: Introduction 11 Overviewmentioning

confidence: 99%

Tropical curves in abelian surfaces I: enumeration of curves passing through points

Blomme¹

2022

Preprint

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This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of Block-Göttsche to get tropical refined invariants.

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Refined count for rational tropical curves in arbitrary dimension

Cited by 4 publications

References 19 publications

Tropical curves in abelian surfaces II: enumeration of curves in linear systems

Tropical curves in abelian surfaces II: enumeration of curves in linear systems

Floor diagrams and enumerative invariants of line bundles over an elliptic curve

Tropical curves in abelian surfaces I: enumeration of curves passing through points

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