Combinatorial algebraic geometry and commutative algebra are two interconnected and active areas of research that use combinatorial tools to approach theoretical, applied, and computational problems in algebra and geometry. These synergies between algebra, geometry, and combinatorics have been useful for developments in these fields, leading to significant advances in toric geometry, enumerative geometry, invariant theory, semigroup algebras, free resolutions, and more. In this session, some recent advancements in combinatorial algebra and geometry will be presented.
La geometría algebraica combinatoria y el álgebra conmutativa son dos áreas de investigación interconectadas y activas que utilizan herramientas combinatorias para abordar problemas teóricos, aplicados y computacionales en álgebra y geometría. Estas sinergias entre álgebra, geometría y combinatoria han sido vitales para desarrollar estos campos, dando lugar a avances significativos en geometría tórica, geometría enumerativa, teoría de invariantes, álgebras de semigrupos, resoluciones libres, y más. En esta sesión, se presentarán algunos avances recientes en álgebra y geometría combinatoria.
05E40
(primary)
05E14; 14N10; 20M14
(secondary)
1.C (1.12)
2.A (1.12)
2.B (1.12)
Systems with multihomogeneous structure can be described using multigraded invariants. We consider multihomogeneous polynomial systems such that the projection to one group of variables is 0-dimensional. In our work, we show that one can rely on the partial regularity region we introduced in a previous article and construct multiplication matrices whose eigenvalues correspond to evaluating the points 0-dimensional projection in linear forms.
Joint work with Matías R. Bender, Laurent Busé and Elias Tsigaridas.
Involutive bases are a special kind of Gröbner bases with additional combinatorial properties. These bases induce free resolutions with nice combinatorial properties. Although these free resolutions are generally far from the minimal, we study the different classes of monomial ideals that lead to minimal resolutions and also to cellular resolutions. We give cellular structures for these cellular resolutions as well as a constructive algorithm that reduces them to the minimal one.
Joint work with Eduardo Sáenz De Cabezón.
The bounded negativity conjecture is an old conjecture that states that there is a lower bound for the self-intersection of reduced and irreducible curves on a smooth complex projective surface that depends only on the surface. In this talk, we present two approaches to this problem by giving bounds that are either linear or quadratic with respect to its intersection with some nef divisor.
Joint work with Carlos Galindo and Francisco Monserrat.
A tensor is identifiable if it admits a unique decomposition as a sum of rank-one tensors. An interesting family consists of $2\times \ldots \times 2$ ($n$ times) tensors since they appear frequently in Bayesian networks, tensor network states, latent-class models, and binary games with multiple players. In this talk, we show that, for every $n\not=3$, a general tensor in $\C^2\otimes \ldots \otimes \C^2 = (\C^2)^{\otimes n}$ is not identifiable.
Joint work with Elisa Postinghel.
Over a field of characteristic zero, De Concini, Eisenbud and Procesi developed a theory of standard monomials, which are a basis for determinantal ideals and provide a tool to study many of their properties, exploiting tableaux combinatorics. In this talk, we present our contribution to a new standard monomial theory in characteristics p modulo a Frobenius power, examining the case p=2. A main feature of this investigation is given by analogs of semistandard Young tableaux in this new context.
Joint work with Teresa Yu.
In this talk, we will see that the minimal number of generators of the prime ideals of Moh, defined in the power series ring in three variables over a field, may change when the characteristic of the field changes. This contradicts a statement of Sally and leaves as an open problem to find families of prime ideals in $K[![x,y,z]!]$ with an unbounded minimal number of generators, when $K$ has characteristic other than zero.
Joint work with Francesc Planas-Vilanova.
Let $\mathcal{C}$ be an integer cone. We say that an affine semigroup $S \subseteq \mathcal{C}$ is a $\mathcal{C}$-semigroup if the complement of $S$ in $\mathcal{C}$ is finite. We present a study of $\mathcal{C}$-semigroups, exploring their properties. Additionally, we develop algorithms for counting $\mathcal{C}$-semigroups with fixed invariants, such as genus and Frobenius elements.
Joint work with José Carlos Rosales and Alberto Vigneron-Tenorio.
We show that a class of bivariate integrals can be seen as generating functions of labeled edge-bicolored graphs. Based on this, we prove an effective algorithm to compute the number of such graphs, and an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex structures. The phase transitions for the asymptotic formulas correspond to critical points of a specific polynomial. We will discuss examples, computation, and conjectures.
Joint work with Michael Borinsky and Maximilian Wiesmann.
The KP equation, a partial differential equation describing nonlinear wave motion, has solutions linked to algebraic curves. Solitons, a special class of solutions, arise from rational nodal curves. Kodama and Williams explored real regular solitons and their connection to totally positive Grassmannians. Building on Abenda and Grinevich's work, I discuss the relationship between real regular solitons, dual graphs of singular curves, Le-graphs, and cells in the totally positive Grassmannian.
Joint work with Simonetta Abenda, Türkü Özlüm Çelik, and Yelena Mandelshtam.
Complete intersection monomial curves (CIMC) illustrate the interplay between curve singularities and the combinatorics of semigroup values from intersection multiplicities with the curve.
This talk will show how a characterization given by Delorme in 1976 helps study the miniversal deformation of CIMCs and provides a decomposition result for its basis.
We will also explore the connection with the moduli space of curves.
Joint work with J.J. Moyano Fernández.
Let $X$ be a rational surface obtained from a Hirzebruch surface by a sequence of blowups centered at closed points. We see conditions which imply, on the one hand, that the efective cone of $X$ is polyhedral and minimally generated and, on the other hand, the finite generation of the Cox ring of $X$. Moreover, we provide a set of generators of the nef cone of $X$ in these cases.
Joint work with C. Galindo and F. Monserrat.
Partially supported by grant PID2022-138906NB-C22 funded by MICIU/AEI/ 10.13039/501100011033 and by ERDF/EU, as well as by Universitat Jaume I, Grants UJI-B2021-02 and GACUJIMB-2023-03.
The Hilbert scheme of points on a variety parametrizes how points behave when they collide. In the case of smooth curves their structure is well understood, whereas for singular curves the general expectation is that is impossible to know. I will report on a work with Javier Sendra Arranz about a class of singular curves for which we describe the irreducible components and their behavior when we increase the number of points and a particularly nice combinatorial description emerging from them.
Joint work with Javier Sendra Arranz.