arXiv subjects

Alejandro Vargas

Publications and source records attributed to Alejandro Vargas.

10 recordsLinked to original sources

arXiv2026

Catalan-many tropical morphisms to trees; Part II: A space and a count

In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus $g$ and degree $d = g/2 + 1$, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number $C_{g/2} = \frac{1}{g/2+1}\binom{g}{g/2}$. In this paper, we construct the tropical counterpart of this universal family: a polyhedral cone complex $\mathcal{G}_{g \to 0, d}^{\mathrm{trop}}$ parametrizing degree-$d$ tropical morphisms from genus-$g$ metric graphs to metric trees. For even $g$ and $d = g/2 + 1$, we prove that the forgetful projection $\Pi \colon \mathcal{G}_{g \to 0, d}^{\mathrm{trop}} \to \mathcal{M}_{g}^{\mathrm{trop}}$ is a branched cover of degree $C_{g/2}$ equipped with natural determinantal multiplicities. We compute this degree by showing that on caterpillars of loops the morphisms are in bijection with ballot sequences, and we establish its global invariance across $\mathcal{M}_{g}^{\mathrm{trop}}$ via a tropical balancing condition across codimension-$1$ walls. Via deformation and path lifting, this yields an effective method to construct Catalan-many gonality-witnessing maps for any generic metric graph, establishing that the tree gonality of any genus-$g$ metric graph is at most $\lceil g/2 \rceil + 1$.

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arXiv2026

About finite differential tropical basis for linear ODE's

We formulate several open questions regarding the tropicalization of linear ODEs, aiming primarily to develop methods for calculating the radius of convergence of their classical solutions. To this aim it is of foremost importance to characterize the classes of equations that admit a finite differential tropical basis, as introduced in (Fink and Toghani, 2022). Our initial exploration examines the second- and third-order cases.

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arXiv2026

Toolkit for the algebraic geometer

In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model spaces via sheaf theory and recover a posteriori the underlying topological spaces that are often present in the construction of such geometric objects. We apply this formalism to semiring schemes and reason why the usual definition of semiring schemes has to be considered as the good approach to the geometry of semirings.

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arXiv2025

Almost-valuative invariants of connected split matroids: The cd-index

We derive a formula for matroid invariants $\Psi$ on a large family of matroids, provided that $\Psi$ is almost-valuative, namely, it satisfies a hyperplane-cut formula. Our primary application is to the cd-index $\Psi_{cd}$ of the base polytope $\mathscr{P}(M)$, a polynomial in two non-commutative variables that compactly encodes the number of face-flags $\mathcal{F} = \{\sigma_1 \subset \dots \subset \sigma_s \}$ with prescribed dimensions $\dim \sigma_i = d_i$. This generalizes recent work by Ferroni and Schr\"oter on the $f$-vector of $\mathscr{P}(M)$, yielding a formula that can be understood as a valuative part plus an error term that surprisingly depends only on modular pairs of cyclic flats. This enables computations requiring only the following data: the evaluations of $\Psi$ on hypersimplices $\Delta_{k,n}$ and cuspidal matroids $\Lambda^{r,h}_{k,n}$; and counts $\lambda(r,h)$ and $\mu(a,b;\alpha,\beta)$ of cyclic flats and modular pairs of cyclic flats in $M$, respectively, satisfying specific rank and cardinality conditions. We compute these for the cd-index, yielding explicit results for sparse paving matroids and rank-2 matroids.

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arXiv2025

Elliptic arrangements of complex multiplication type

We provide a natural definition of an elliptic arrangement, extending the classical framework to an elliptic curve E with complex multiplication. We analyse the intersections of elements of the arrangement and their connected components as End(E)-modules. Furthermore, we prove that the combinatorial data of elliptic arrangements define both an arithmetic matroid and a matroid over the ring End(E). In this way, we obtain a class of arithmetic matroids that is different from the class of arithmetic matroids realizable via toric arrangements. Finally, we show that the Euler characteristic of the complement is an evaluation of the arithmetic Tutte polynomial.

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arXiv2023

Combinatorics of higher-dimensional tropical covers

We develop a combinatorial framework to study certain polyhedral maps which are higher-dimensional analogues of tropical covers between metric graphs. Under a mild combinatorial assumption, we show that a map satisfies the so-called balancing condition if and only if it is an indexed branched cover, i.e.~locally over connected sets the count with multiplicity of points in every fibre is a constant, which in particular gives a well-defined global degree when the target is connected. Given a balanced map $(\Sigma, m_\Sigma) \to \Delta$, we lift several connectivity properties of $\Delta$ to~$\Sigma$. Using these lifting results we determine whether a multiplicity $m_{\mathcal U}$ that is defined only on the interiors of maximal cells of $\Sigma$ can be extended to all $\Sigma$ in a balanced manner. This relies on a strong connectivity assumption; we give a counterexample when this is missing.

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arXiv2023

Buildings, valuated matroids, and tropical linear spaces

Affine Bruhat--Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of $\mathrm{PGL}$ parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings $\iota\colon\mathbb{P}^r\hookrightarrow\mathbb{P}^n$ and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.

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arXiv2023

One-quasihomomorphisms from the integers into symmetric matrices

A function $f$ from $\mathbb{Z}$ to the symmetric matrices over an arbitrary field $K$ of characteristic $0$ is a $1$-quasihomomorphism if the matrix $f(x+y) - f(x) - f(y)$ has rank at most $1$ for all $x,y \in \mathbb{Z}$. We show that any such $1$-quasihomomorphism has distance at most $2$ from an actual group homomorphism. This gives a positive answer to a special case of a problem posed by Kazhdan and Ziegler.

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arXiv2019

A short note on Cayley-Salmon equations

A Cayley-Salmon equation for a smooth cubic surface $S$ in $\mathbb P^3$ is an expression of the form $l_1l_2l_3 - m_1m_2m_3 = 0$ such that the zero set is $S$ and $l_i$, $m_j$ are homogeneous linear forms. This expression was first used by Cayley and Salmon to study the incidence relations of the 27 lines on $S$. There are 120 essentially distinct Cayley-Salmon equations for $S$. In this note we give an exposition of a classical proof of this fact. We illustrate the explicit calculation to obtain these equations and we apply it to Clebsch surface and to the octanomial model. Finally we show that these $120$ Cayley-Salmon equations can be directly computed using recent work by Cueto and Deopurkar.

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arXiv2019

Catalan-many tropical morphisms to trees; Part I: Constructions

We investigate the tree gonality of a genus-$g$ metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this number is at most $\lceil g/2 \rceil + 1$, a fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, the tropical morphism which realizes the bound belongs to a family of tropical morphisms that is pure of dimension $3g-3$ and that has a generically finite-to-one map onto the moduli space of genus-$g$ metric graphs. Our methods focus on the study of such families. This is part I in a series of two papers: in part I we fix the combinatorial type of the metric graph, while in part II we vary the combinatorial type and show that the number of tropical morphisms, counted with suitable multiplicities, is the same Catalan number that counts morphisms from a genus-$g$ curve to the projective line.

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