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Hannah Markwig

Publications and source records attributed to Hannah Markwig.

At least 19 recordsLinked to original sources
arXiv2026

Tropicalization and degeneration of short flag varieties and their fibers

Tropicalizations of Grassmannians resp. flag varieties parametrizing (flags of) linear spaces have been studied intensely and reveal rich combinatorial structure. Linear degenerations of flag varieties provide ways of interpolating between products of Grassmannians and flag varieties. Tropical versions of these have been introduced recently, but their combinatorial description and computation remains a challenge. We consider a particular case which allows the use of matroidal methods, namely linear degenerate short flag varieties parametrizing tuples of linear spaces such that a projection of one is contained in the other. Their fibers turn out to be linear spaces themselves under mild assumptions, which allows a comprehensive combinatorial description of their tropicalizations, and accordingly, of tropicalizations of linear degenerate short flag varieties.

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arXiv2026

Quadratically Enriched Plane Curve Counting via Tropical Geometry

We prove that the quadratically enriched count of rational curves in a smooth toric del Pezzo surface passing through $k$-rational points and pairs of conjugate points in quadratic field extensions $k\subset k(\sqrt{d_i})$ can be determined by counting certain tropical stable maps through vertically stretched point conditions with a suitable multiplicity. Building on the floor diagram technique in tropical geometry, we provide an algorithm to compute these numbers. Our tropical algorithm computes not only these new quadratically enriched enumerative invariants, but simultaneously also the complex Gromov-Witten invariant, the real Welschinger invariant counting curves satisfying real point conditions only, the real Welschinger invariant of curves satisfying pairs of complex conjugate and real point conditions, and the quadratically enriched count of curves satisfying $k$-rational point conditions.

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arXiv2026

Tropical methods for building real space sextics with totally real tritangent planes

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.

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arXiv2026

A lifting partition theorem for tropical tritangent classes to smooth space sextic curves

The set of tritangent planes to smooth tropical space sextic curves has 15 connected components, recording continuous displacements of planes preserving the tritangency condition. These 15 tritangent classes are polyhedral complexes in $\mathbf{R}^3$, and each of them contains the tropicalization of precisely eight tritangent planes to any smooth space sextic curve with the given tropicalization. Prior joint work of the authors with Len confirms that each tropical tritangent plane has 0, 1, 2, 4 or 8 lifts to classical tritangent planes defined over the algebraic closure of the field over which the original algebraic curve is defined. Our main theorem states that when the input classical curve is generic, then only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes. Furthermore, we show that these partitions are completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type.

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arXiv2026

Trigonal and embedded tropical curves of low genus

In algebraic geometry, trigonal curves can always be embedded into Hirzebruch surfaces. In tropical geometry, the notion of trigonality does not have a unique translation. We focus on the characterization in terms of the existence of a degree 3 morphism to a line, and discuss relations to possible embeddings into $\mathbb R^2$ reflecting an embedding into a Hirzebruch surface. Our results can be divided into three parts: for tropical curves of low genus 3 and 4, we discuss the relation between a trigonal morphism and an embedding dual to the polygon of a Hirzebruch surface, building on works on embeddings of hyperelliptic tropical curves and curves of low genus. We compare obstructions for embeddings with obstructions for the existence of a degree 3 morphism to a line. Finally, we showcase examples where a non-smooth embedding can be unfolded to reflect certain features of a degree 3 morphism to a line.

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arXiv2026

Tropical Methods for Counting Plane Curves -- Complex, Real and Quadratically Enriched

Since the first famous correspondence theorem by Mikhalkin appeared in 2005, tropical geometry has allowed a parallel treatment of real and complex counting problems. A prime example are the genus 0 Gromov-Witten invariants of the plane which count rational plane curves of degree d satisfying point conditions and their real counterpart, the Welschinger invariants, which both can be determined using tropical methods. Remarkably, the tropical computation of the two types of invariants works entirely in parallel. Recently, quadratically enriched enumerative geometry enables us to combine such real and complex counts under one roof, providing a simultaneous approach which can also be used for counts over other fields. Tropical geometry is a successful tool for the study and computation of such quadratically enriched enumerative invariants, too. In this survey, we provide an overview of tropical methods for plane curve counting problems over the real and complex numbers, and the new quadratically enriched counts.

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arXiv2025

One part leaky covers

In our previous work [CMS24] we defined a new class of enumerative invariants called $k$-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and $k$-leaky double Hurwitz numbers. Here, we focus on the one-part version of these numbers, i.e.\ when the positive ramification profile is $(d)$. We derive recursions and use them to produce explicit formulas and structure results for some infinite families of these numbers.

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arXiv2025

A refined twist on Hurwitz numbers

We introduce a two-parameter refinement of the Jucys-Murphy theory, that we call the CJT-refinement, unifying Schur, zonal, and, conjecturally, Jack actions of the ring of symmetric functions on the Fock space. Applications of this formalism include a partial resolution of a recent conjecture of Coulter-Do, as well as cut-and-join recursion for $b$-Hurwitz numbers. The cut-and-join equations enable the derivation of the tropicalization of $b$--Hurwitz numbers. We also provide a first application of this tropical interpretation by answering an open problem of Chapuy-Dołęga on the polynomial structure of $b$-Hurwitz numbers.

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arXiv2025

Pluricanonical cycles and tropical covers

We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

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arXiv2024

Tropical twisted Hurwitz numbers for elliptic curves

Hurwitz numbers enumerate branched morphisms between Riemann surfaces. For a fixed elliptic target, Hurwitz numbers are intimately related to mirror symmetry following work of Dijkgraaf. In recent work of Chapuy and Dolega a new variant of Hurwitz numbers with fixed genus $0$ target was introduced that includes maps between between non-orientiable surfaces. These numbers are called $b$-Hurwitz numbers and are polynomials in a parameter $b$ which measures the non-orientability of the involved maps. An interpretation in terms of factorisations of $b$-Hurwitz numbers for $b=1$, so-called twisted Hurwitz numbers, was found in work of Burman and Fesler. In previous work, the authors derived a tropical geometry interpretation of these numbers. In this paper, we introduce a natural generalisation of twisted Hurwitz numbers with elliptic targets within the framework of symmetric groups. We derive a tropical interpretation of these invariants, relate them to Feynman integrals and derive an expression as a matrix element of an operator in the bosonic Fock space.

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arXiv2024

Arithmetic Counts of Tropical Plane Curves and Their Properties

Recently, the first and third author proved a correspondence theorem which recovers the Levine-Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study properties of the arithmetic count of plane tropical curves satisfying point conditions. We prove that this count is independent of the configuration of point conditions. Moreover, a Caporaso-Harris formula for the arithmetic count of plane tropical curves is obtained by moving one point to the very left. Repeating this process until all point conditions are stretched, we obtain an enriched count of floor diagrams which coincides with the tropical count. Finally, we prove polynomiality properties for the arithmetic counts using floor diagrams.

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arXiv2024

$k$-leaky double Hurwitz descendants

We define a new class of enumerative invariants called $k$-leaky double Hurwitz descendants, generalizing both descendant integrals of double ramification cycles and the $k$-leaky double Hurwitz numbers introduced in previous work of Cavalieri, Markwig and Ranganathan. These numbers are defined as intersection numbers of the logarithmic DR cycle against $ψ$-classes and logarithmic classes coming from piecewise polynomials encoding fixed branch point conditions. We give a tropical graph sum formula for these new invariants, allowing us to show their piecewise polynomiality and a wall-crossing formula in genus zero. We also prove that in genus zero the invariants are always non-negative and give a complete classification of the cases where they vanish.

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arXiv2023

Twisted Hurwitz numbers: Tropical and polynomial structures

Hurwitz numbers count covers of curves satisfying fixed ramification data. Via monodromy representation, this counting problem can be transformed to a problem of counting factorizations in the symmetric group. This and other beautiful connections make Hurwitz numbers a longstanding active research topic. In recent work Chapuy and Dolęga, a new enumerative invariant called b-Hurwitz number was introduced, which enumerates non-orientable branched coverings. For b=1, we obtain twisted Hurwitz numbers which were linked to surgery theory in work of Burman and Fesler and admit a representation as factorisations in the symmetric group. In this paper, we derive a tropical interperetation of twisted Hurwitz numbers in terms of tropical covers and study their polynomial structure.

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arXiv2023

Algorithms for Gromov-Witten Invariants of Elliptic Curves

We present an enhanced algorithm for exploring mirror symmetry for elliptic curves through the correspondence of algebraic and tropical geometry, focusing on Gromov-Witten invariants of elliptic curves and, in particular, Hurwitz numbers. We present a new highly efficient algorithm for computing generating series for these numbers. We have implemented the algorithm both using Singular and OSCAR. The implementations outperform by far the current method provided in Singular. The OSCAR implementation, benefiting in particular from just-in-time compilation, again by far outperforms the implementation of the new algorithm in Singular. This advancement in computing the Gromov-Witten invariants facilitates a study of number theoretic and geometric properties of the generating series, including quasi-modularity and homogeneity.

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arXiv2023

Faithful tropicalization of hyperelliptic curves

We provide explicit faithful re-embeddings for all hyperelliptic curves of genus at most three and an algorithmic way to construct them. Both in the faithful tropicalization algorithm and the proofs of correctness, we showcase OSCAR-methods for commutative algebra, polyhedral and tropical geometry.

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arXiv2023

Combinatorics and real lifts of bitangents to tropical quartic curves

Smooth algebraic plane quartics over algebraically closed fields have 28 bitangent lines. Their tropical counterparts often have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic. We show such classes determine tropically convex sets and provide a complete combinatorial classification of such objects into 41 types (up to symmetry). The occurrence of a given class is determined by both the combinatorial type and the metric structure of the input tropical plane quartic. We use this result to provide explicit sign-rules to obtain real lifts for each tropical bitangent class, and confirm that each one has either zero or exactly four real lifts, as previously conjectured by Len and the second author. Furthermore, such real lifts are always totally-real.

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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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arXiv2023

Avoidance loci and tropicalizations of real bitangents to plane quartics

We compare two partitions of real bitangents to smooth plane quartics into sets of 4: one coming from the closures of connected components of the avoidance locus and another coming from tropical geometry. When both are defined, we use the Tarski principle for real closed fields in combination with the topology of real plane quartics and the tropical geometry of bitangents and theta characteristics to show that they coincide.

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