arXiv subjects

Thomas Blomme

Publications and source records attributed to Thomas Blomme.

At least 19 recordsLinked to original sources

arXiv2026

A correlated refinement of the double double ramification cycle

Given a family of semi-stable curves together with two degree 0 line bundles, the double double ramification cycle measures the locus where both line bundles are trivial on the fibers. When the two line bundles come equipped with natural roots, we provide a refinement of the DDR-class using the Weil pairing of the roots. We prove that the refined classes satisfy a multiple cover formula analogous to the one for correlated invariants of projective bundles on elliptic curves proved in [BC25b]. As a consequence, we prove that log-GW invariants of toric surfaces can be refined taking into account the position of the points mapped to the toric boundary, and that these refined invariants also satisfy a multiple cover formula; the latter is as a variation of the N. Takahashi conjecture for genus zero maximal contact curves for P2 relative a smooth elliptic curve E.

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arXiv2026

Refined invariants for Abelian surfaces: between polynomiality and modularity

Tropical refined invariants for toric surfaces, introduced Block and G{\"o}ttsche, are obtained couting tropical curves with a Laurent polynomial multiplicity. Brugall{\'e} and Jaramillo-Puentes then exhibited a polynomial behavior of the coefficients of this Laurent polynomial, seen as function on the curve degree. The authors provided explicit formula for small genus, involving quasi-modular forms. Inspired by the toric setting, the first-named author defined refined invariants for abelian surfaces and extended the polynomiality result. In this paper, we further study this regularity for abelian surfaces, providing explicit formulas involving quasi-modular forms. This resonates with the small genus cases of the toric setting.

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arXiv2025

Multiple cover formulas for abelian surfaces via correlated invariants

We prove the multiple cover formula conjecture for abelian surfaces for a large class of insertions, including all stationary invariants. The proof uses the reduced degeneration formula expressing the invariants in terms of the correlated Gromov--Witten invariants previously introduced by the authors.

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arXiv2025

Correlated double ramification cycle formula

We prove a refinement of Pixton's formula for the double ramification cycle with target variety which takes into account the correlator of a rubber map previously introduced by the authors. To do so, we need to: reinterpret the correlator in terms of (logarithmic) roots of the trivial line bundle; refine the natural stratification of the boundary of moduli of maps to keep track of how torsion line bundles on the target variety pull-back. We apply the refined DR cycle formula to compute 0-correlated invariants for genus g curve with points and a $\lambda$-class insertion.

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arXiv2025

On the double tangent of projective closed curves

We generalize a previous result by Fabricius-Bjerre from curves in $\mathbb R^2$ to curves in $\mathbb R P^2$. Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson-Vogt and proves its positivity, conjectured by Larson-Vogt. Our method is not specific to quartics and applies to algebraic curves of any degree.

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arXiv2025

Asymptotic computations of tropical refined invariants in genus 0 and 1

Block and Göttsche introduced a Laurent polynomial multiplicity to count tropical curves. Itenberg and Mikhalkin then showed that this multiplicity leads to invariant counts called tropical refined invariants. Recently, Brugallé and Jaramillo-Puentes studied the polynomiality properties of the coefficients of these invariants and showed that for fixed genus g, the coefficients ultimately coincide with polynomials in the homology class of the curves we look at. We call the generating series of these polynomials asymptotic refined invariant. In genus 0, the asymptotic refined invariant has been computed by the second author in the h-transverse case. In this paper, we give a new proof of the formula for the asymptotic refined invariant for g = 0 using variations on the floor diagram algorithm. This technique allows also to compute the asymptotic refined invariant for g = 1. The result exhibits surprising regularity properties related to the generating series of partition numbers and quasi-modular forms.

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arXiv2025

A short proof of the multiple cover formula for point insertions

A few years ago, G. Oberdieck conjectured a multiple cover fomula that determines the number of curves of fixed genus and degree passing through a configuration of points in an abelian surface. This formula was proved by the author using tropical techniques and Nishinou's correspondence theorem. Using the same techniques, we give a much shorter proof of the multiple cover formula for point insertions, relying on the same geometrical idea, but avoiding any kind of tropical enumeration.

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arXiv2024

Correlated Gromov-Witten Invariants

We introduce a geometric refinement of Gromov-Witten invariants for $\mathbb P^1$-bundles relative to the natural fiberwise boundary structure. We call these refined invariant correlated Gromov-Witten invariants. Furthermore, we prove a refinement of the degeneration formula keeping track of the correlation. Finally, combining certain invariance properties of the correlated invariant, a local computation and the refined degeneration formula we follow floor diagram techniques to prove regularity results for the generating series of the invariants in the case of $\mathbb P^1$-bundles over elliptic curves. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces.

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arXiv2024

Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas

This paper is the third installment in a series of papers devoted to the computation of enumerative invariants of abelian surfaces through the tropical approach. We develop a pearl diagram algorithm similar to the floor diagram algorithm used in toric surfaces that concretely solves the tropical problem. These diagrams can be used to prove specific cases of Oberdieck's multiple cover formula that reduce the computation of invariants for non-primitive classes to the primitive case, getting rid of all diagram considerations and providing short explicit formulas. The latter can be used to prove the quasi-modularity of generating series of classical invariants, and the polynomiality of coefficients of fixed codegree in the refined invariants.

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arXiv2024

Bitangents of real algebraic curves: signed count and constructions

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic real algebraic curve of even degree $d$ has at least $\frac{d(d-2)}{2}$ real bitangents. Next we explain how to locate (real) bitangents of a (real) perturbation of a multiple (real) conic in $\mathbb{C}P^2$. As main applications, we exhibit a real sextic with a total of $318$ real bitangents and 6 complex ones, and perform asymptotical constructions that give the best, to our knowledge, number of real bitangents of real algebraic plane curves of a given degree.

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arXiv2024

Gromov-Witten Invariants of Bielliptic Surfaces

Bielliptic surfaces appear as quotient of a product of two elliptic curves and were classified by Bagnera-Franchis. We give a concrete way of computing their GW-invariants with point insertions using a floor diagram algorithm. Using the latter, we are able to prove the quasi-modularity of their generating series by relating them to generating series of graphs for which we also prove quasi-modularity results. We propose a refinement of these invariants by inserting a λ-class in the considered GW-invariants.

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arXiv2023

Tropical Möbius strips and ruled surfaces

We consider the enumeration of tropical curves in Möbius strips for two different lattice structures and relate them to the enumeration of curves in two rational ruled surfaces over a complex elliptic curve. Using this correspondence, we prove regularity results such as the piecewise quasi-polynomiality of relative invariants and the quasi-modularity of their generating series.

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arXiv2023

Tropical descendant invariants with line constraints

Via correspondence theorems, rational log Gromov--Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalized) lattice path counts and floor diagram techniques. So far, the cases for which there exist algorithms do not extend to non-stationary rational descendant log Gromov--Witten invariants, i.e.\ those where Psi-conditions do not have to be matched up with the evaluation of a point. The case of rational descendant log Gromov--Witten invariants satisfying point conditions (without Psi-conditions) and one Psi-condition of any power combined with a line plays a particularly important role, since it shows up in mirror symmetry as coefficients of the $J$-function. We provide recursive formulas to compute those numbers via tropical methods. Our method is inspired by the tropical proof of the WDVV equations. We also extend our study to counts involving two lines, both paired up with a Psi-condition, appearing with power one.

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arXiv2022

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

We use the tropical geometry approach to compute absolute and relative Gromov-Witten invariants of complex surfaces which are $\CC P^1$-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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arXiv2022

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

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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arXiv2022

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

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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arXiv2021

Refined count of real oriented rational curves

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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arXiv2021

Refined count for rational tropical curves in arbitrary dimension

In this paper we introduce a refined multiplicity for rational tropical curves in arbitrary dimension, which generalizes the refined multiplicity introduced by F. Block and L. Göttsche in arXiv:1407.2901 . We then prove an invariance statement for the count of rational tropical curves in several enumerative problems using this new refined multiplicity. This leads to the definition of Block-Göttsche polynomials in any dimension.

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