arXiv subjects

Steffen Marcus

Publications and source records attributed to Steffen Marcus.

14 recordsLinked to original sources

arXiv2024

Tropical subrepresentations of the boolean regular representation in low dimension

We study two dimensional and three dimensional tropical subrepresentations of the regular representation $\mathbb{B}[G]$ of a finite group over the tropical booleans, utilizing the theory of group representations over a fixed idempotent semifield as developed by Giansiracusa--Manaker. In dimension two we completely classify all two dimensional tropical subrepresentations of $\mathbb{B}[G]$, provide an explicit characterization for the set of bases of the corresponding matroids, and show an equivalence with the subgroups of $G$. In dimension three we show such an equivalence no longer holds. Towards a classification in dimension three we give a collection of tropical subrepresentations corresponding to subgroups of index 2, and we show that in the special case of finite cyclic groups, one can find three dimensional tropical subrepresentations that do not correspond to subgroups in a similar way.

math.RT↗

arXiv2024

Tropical pseudostable curves

We study the tropical version of the contraction morphism $\mathcal{T}$ between moduli spaces of stable and pseudostable curves. By promoting $\mathcal{T}$ to a logarithmic morphism, we obtain a piecewise linear function between the generalized cone complexes parameterizing tropical stable and pseudostable curves. The ray corresponding to the contracted divisor $\delta_1$ is not contracted to the cone point but mapped onto a ray of $\mathcal{M}_{g,n}^{{\rm trop}, {\rm ps}}$, with a slope reflecting the geometry of the desingularization of a plane cusp. We explore in detail the situation of $g=1$, where the tautological geometry of both spaces is fully described by piecewise polynomial functions on the tropical moduli spaces.

math.AG↗

arXiv2021

Logarithmic compactification of the Abel-Jacobi section

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry, we describe a modular modification of the moduli space of curves over which the Abel-Jacobi map extends. We also describe the attendant deformation theory and virtual fundamental class of this moduli space. This recovers the double ramification cycle, as well as variants associated to differentials.

math.AG↗

arXiv2021

Boundedness of the space of stable logarithmic maps

We prove that the moduli space of stable logarithmic maps with fixed numerical invariants, from logarithmic curves to a fixed projective target logarithmic scheme with fine and saturated logarithmic structure, is a proper algebraic stack. This was previously known only with further restrictions on the logarithmic structure of the target.

math.AG↗

arXiv2019

Topology of tropical moduli of weighted stable curves

The moduli space $Δ_{g,w}$ of tropical $w$-weighted stable curves of volume $1$ is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of $w$-weighted stable curves. If at least two of the weights are $1$, we prove that $Δ_{0,w}$ is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space $Δ_{0,w}$ is disconnected and for which the fundamental group of $Δ_{0,w}$ has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri-Hampe-Markwig-Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus $0$ and $1$, and leverage this to extend several of our genus $0$ results to the spaces $Δ_{1,w}$.

math.CO↗

arXiv2018

Clutching and gluing in tropical and logarithmic geometry

The classical clutching and gluing maps between the moduli stacks of stable marked algebraic curves are not logarithmic, i.e. they do not induce morphisms over the category of logarithmic schemes, since they factor through the boundary. Using insight from tropical geometry, we enrich the category of logarithmic schemes to include so-called sub-logarithmic morphisms and show that the clutching and gluing maps are naturally sub-logarithmic. Building on the recent framework developed by Cavalieri, Chan, Wise, and the third author, we further develop a stack-theoretic counterpart of these maps in the tropical world and show that the resulting maps naturally commute with the process of tropicalization.

math.AG↗

arXiv2017

Two-vertex generators of Jacobians of graphs

We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.

math.CO↗

arXiv2015

Skeletons and fans of logarithmic structures

We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities.

math.AG↗

arXiv2013

Stable maps to rational curves and the relative Jacobian

We consider two cycles on the moduli space of compact type curves and prove that they coincide. The first is defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized rational curve, while the second is obtained as the intersection of the Abel section of the universal Jacobian with the zero section. Our comparison extends results of Cavalieri-Marcus-Wise where the same identity was proved over on the locus of rational tails curves.

math.AG↗

arXiv2013

A geometric perspective on the piecewise polynomiality of double Hurwitz numbers

We describe double Hurwitz numbers as intersection numbers on the moduli space of curves. Assuming polynomiality of the Double Ramification Cycle (which is known in genera 0 and 1), our formula explains the polynomiality in chambers of double Hurwitz numbers, and the wall crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle.

math.AG↗

arXiv2013

Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations

We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: Jun Li's original approach, Bumsig Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory without expansions due to Gross-Siebert and Abramovich-Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov-Witten invariants associated to all four of these theories are identical.

math.AG↗

arXiv2011

Polynomial families of tautological classes on $\mathcal{M}_{g,n}^{rt}$

We study classes $P_{g,T}(α;β)$ on the moduli space of stable, genus g curves with rational tails defined by pushing forward the virtual fundamental classes of spaces of relative stable maps to an unparameterized projective line. A comparison with classes $Q_{g,T}$ arising from sections of the universal Jacobian shows the classes $P_{g,T}(α;β)$ are polynomial in the parts of the partitions indexing the special ramification data. Virtual localization on moduli spaces of relative stable maps gives sufficient relations to compute the coefficients of these polynomials in various cases.

math.AG↗

arXiv2010

The Evaluation Space of Logarithmic Stable Maps

The evaluation stack for minimal logarithmic stable maps is constructed, parameterizing families of standard log points in the target log scheme. This construction provides the ingredients necessary to define appropriate evaluation maps for minimal log stable maps and establish the logarithmic Gromov-Witten theory of a log-smooth Deligne- Faltings log scheme.

math.AG↗

arXiv2010

Very Twisted Stable Maps

Let X be a smooth projective Deligne-Mumford stack over an algebraically closed field k of characteristic 0. In this paper we construct the moduli stack of very twisted stable maps, extending the notion of twisted stable maps by Abramovich and Vistoli to allow for generic stabilizers on the source curves. We also consider the Gromov-Witten theory given by this construction.

math.AG↗