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Alex Abreu

Publications and source records attributed to Alex Abreu.

At least 19 recordsLinked to original sources
arXiv2024

Counting points on Hessenberg Varieties over finite fields

We give a counting formula in terms of modified Hall-Littlewood polynomials and the chromatic quasisymmetric function for the number of points on an arbitrary Hessenberg variety over a finite field. As a consequence, we express the Poincar\'e polynomials of complex Hessenberg varieties in terms of a Hall scalar product involving the symmetric functions above. We use these results to give a new proof of a combinatorial formula for the modified Hall-Littlewood polynomials.

math.CO↗
arXiv2023

A Torelli theorem for graphs via quasistable divisors

The Torelli theorem establishes that the Jacobian of a smooth projective curve, together with the polarization provided by the theta divisor, fully characterizes the curve. In the case of nodal curves, there exists a concept known as fine compactified Jacobian. The fine compactified Jacobian of a curve comes with a natural stratification that can be regarded as a poset. Furthermore, this poset is entirely determined by the dual graph of the curve and is referred to as the poset of quasistable divisors on the graph. We present a combinatorial version of the Torelli theorem, which demonstrates that the poset of quasistable divisors of a graph completely determines the biconnected components of the graph (up to contracting separating edges). Moreover, we achieve a natural extension of this theorem to tropical curves.

math.CO↗
arXiv2023

Splitting the cohomology of Hessenberg varieties and e-positivity of chromatic symmetric functions

For each indifference graph, there is an associated regular semisimple Hessenberg variety, whose cohomology recovers the chromatic symmetric function of the graph. The decomposition theorem applied to the forgetful map from the regular semisimple Hessenberg variety to the projective space describes the cohomology of the Hessenberg variety as a sum of smaller pieces. We give a combinatorial description of the Frobenius character of each piece. This provides a generalization of the symmetric functions attached to Stanley's local h-polynomials of the permutahedral variety to any Hessenberg variety. As a consequence, we can prove that the coefficient of $e_{\lambda}$, where $\lambda$ is any partition of length 2, in the e-expansion of the chromatic symmetric function of any indifference graph is non-negative.

math.AG↗
arXiv2023

Wall-crossing of universal Brill-Noether classes

We give an explicit graph formula, in terms of decorated boundary strata classes, for the wall-crossing of universal Brill-Noether classes. More precisely, fix n>0 and d<g , and two stability conditions \phi^-, \phi^+ for degree d compactified universal (over the moduli space of stable n-pointed curves of genus g) Jacobians that lie on opposite sides of a stability hyperplane. Our main result is a formula for the difference between the Brill-Noether classes, compared via the pullback along the (rational) identity map. The calculation involves constructing a resolution of the identity map by means of subsequent blow-ups.

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arXiv2022

Parabolic Lusztig varieties and chromatic symmetric functions

The characters of Kazhdan--Lusztig elements of the Hecke algebra over $S_n$ (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of certain subvarieties of the flag variety. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of $S_n$. An explicit characterization of such representations would provide a recursive formula for the computation of such characters/chromatic symmetric functions, which could settle Haiman's conjecture about the positivity of the monomial characters of Kazhdan--Lusztig elements and Stanley--Stembridge conjecture about $e$-positivity of chromatic symmetric function of indifference graphs. We also find a connection between the character of certain homology groups of subvarieties of the partial flag varieties and the Grojnowski--Haiman hybrid basis of the Hecke algebra.

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arXiv2022

An update on Haiman's conjectures

We revisit Haiman's conjecture on the relations between characters of Kazdhan-Lusztig basis elements of the Hecke algebra over the symmetric group. The conjecture asserts that, for purposes of character evaluation, any Kazhdan-Lusztig basis element is reducible to a sum of the simplest possible ones (those associated to so-called codominant permutations). When the basis element is associated to a smooth permutation, we are able to give a geometric proof of this conjecture. On the other hand, if the permutation is singular, we provide a counterexample.

math.AG↗
arXiv2022

A geometric approach to characters of Hecke algebras

To any element of a connected, simply connected, semisimple complex algebraic group G and a choice of an element of the corresponding Weyl group there is an associated Lusztig variety. When the element of G is regular semisimple, the corresponding variety carries an action of the Weyl group on its (equivariant) intersection cohomology. From this action, we recover the induced characters of an element of the Kazhdan-Lusztig basis of the corresponding Hecke algebra. In type A, we prove a more precise statement: that the Frobenius character of this action is precisely the symmetric function given by the characters of a Kazhdan-Lusztig basis element. The main idea is to find celular decompositions of desingularizations of these varieties and apply the Brosnan-Chow palindromicity criterion for determining when the local invariant cycle map is an isomorphism. This recovers some results of Lusztig about character sheaves and gives a generalization of the Brosnan-Chow solution to the Sharesian-Wachs conjecture to non-codominant permutations, where singularities are involved. We also review the connections between Immanants, Hecke algebras, and Chromatic quasisymmetric functions of indifference graphs.

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arXiv2022

On moduli spaces of roots in algebraic and tropical geometry

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.

math.AG↗
arXiv2022

Degree-2 Abel maps and hyperelleptic curves

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an application, we characterize when the (symmetrized) degree-2 Abel map is not injective, a property that, for a smooth curve, is equivalent to the curve being hyperelliptic.

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arXiv2020

The moduli space of quasistable spin curves

We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.

math.AG↗
arXiv2020

The resolution of the universal Abel map via tropical geometry and applications

Let $g$ and $n$ be nonnegative integers and $\mathcal A=(a_0,\dots,a_n)$ a sequence of $n+1$ integers summing up to $d$. Let $\overline{\mathcal M}_{g,n+1}$ be the moduli space of $(n+1)$-pointed stable curves of genus $g$ and $\overline{\mathcal J}_{μ,g}\rightarrow \overline{\mathcal M}_{g,1}$ be the Esteves' universal Jacobian, where $μ$ is a universal genus-$g$ polarization of degree $d$. We give an explicit resolution of the universal Abel map $α_{\mathcal A,μ}\colon \overline{\mathcal M}_{g,n+1}\dashrightarrow \overline{\mathcal J}_{μ,g}$, taking a pointed curve $(X,p_0,\dots,p_n)$ to $\mathcal{O}_X(\sum_{0\le i\le n} a_ip_i)$. The blowup of $\overline{\mathcal M}_{g,n+1}$ giving rise to the resolution is inspired by the resolution of the tropical analogue of the map $α_{\mathcal A,μ}$ (in the category of generalized cone complexes). As an application, we describe the double ramification cycle in terms of the universal sheaf inducing the resolution of the map $α_{\mathcal A,μ}$.

math.AG↗
arXiv2020

Abel Maps for nodal curves via tropical geometry

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.

math.AG↗
arXiv2020

Chromatic symmetric functions from the modular law

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced by Guay-Paquet. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain $q$-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.

math.CO↗
arXiv2020

A symmetric function of increasing forests

For an indifference graph $G$ we define a symmetric function of increasing spanning forests of $G$. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular LLT polynomials. As a consequence we give a combinatorial interpretation of the coefficients of the LLT polynomial in the elementary basis (up to a factor of a power of $(q-1)$), strengthening the description given by Alexandersson and Sulzgruber.

math.CO↗
arXiv2019

A universal tropical Jacobian over $M_g^{\trop}$

We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus $g$. This space parametrizes equivalence classes of tropical curves of genus $g$ together with a $μ$-polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decompositions.

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arXiv2019

The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian

For each universal genus-$g$ polarization $μ$ of degree $d$, we construct a universal tropical Jacobian $J_{μ,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{μ,g}^{trop}$. In particular, we prove that the natural compactification of $J_{μ,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.

math.AG↗
arXiv2016

Enriched curves and their tropical counterpart

In her Ph.D. thesis, Mainò introduced the notion of enriched structure on stable curves and constructed their moduli space. In this paper we give a tropical notion of enriched structure on tropical curves and construct a moduli space parametrizing these objects. Moreover, we use this construction to give a toric description of the scheme parametrizing enriched structures on a fixed stable curve.

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