arXiv subjects

Cristhian Garay

Publications and source records attributed to Cristhian Garay.

6 recordsLinked to original sources

arXiv2026

A Real Shafarevich Conjecture for Universal Covers

The classical Shafarevich conjecture predicts that the universal cover of a complex smooth projective variety $X$ is holomorphically convex. In this paper, we propose a refinement of this conjecture for varieties defined over the reals. In order to do this, we introduce the notions of real holomorphic convexity and transverse holomorphic convexity to capture the geometric differences dictated by the real locus $X(\mathbb{R})$ of $X$. Specifically, we conjecture that the universal cover is real holomorphically convex when $X(\mathbb{R}) \neq \emptyset$, and dianalytic holomorphically convex when $X(\mathbb{R}) = \emptyset$. We prove this refined conjecture in two main cases: when $X$ is a curve, and when the fundamental group of $X$ is nilpotent.

math.AG↗

arXiv2024

Idempotentization of Affine Schemes and Sheaves

In this article, we introduce the idempotentization process, which bears some philosophical and mathematical similarities with modern analytification and tropicalization. Idempotentization associates to any affine scheme an idempotent version of itself with respect to a fixed covering by distinguished affine open subschemes. Once this cover is fixed, we can functorially associate a Zariski sheaf of rings or modules to a sheaf of idempotent semiring or a sheaf of idempotent semimodules. We show that idempotentization is independent of the chosen cover and in the Noetherian case, the idempotentization of the structure sheaf recovers the global sections. Underlying our formalism is a combinatorial reflection of lattices of subobjects of ordered-theoretic objects seen as lattices coming from commutative algebra. This has topological consequences for the semiring of subtractive ideals of a commutative semiring $S$: On one hand, it is a topological retract of the semiring of congruence relations of $S$ for the coarse lower topology. On the other hand, it is a topological retract of the semiring of ideals of $S$ for the coarse upper topology.

math.AG↗

arXiv2024

Bitangents of real algebraic curves: signed count and constructions

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic real algebraic curve of even degree $d$ has at least $\frac{d(d-2)}{2}$ real bitangents. Next we explain how to locate (real) bitangents of a (real) perturbation of a multiple (real) conic in $\mathbb{C}P^2$. As main applications, we exhibit a real sextic with a total of $318$ real bitangents and 6 complex ones, and perform asymptotical constructions that give the best, to our knowledge, number of real bitangents of real algebraic plane curves of a given degree.

math.AG↗

arXiv2023

On $T$-invariant subvarieties of symplectic Grassmannians and representability of rank $2$ symplectic matroids over ${\mathbb C}$

For the symplectic Grassmannian $\text{SpG}(2,2n)$ of $2$-dimensional isotropic subspaces in a $2n$-dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an $n$-dimensional torus $T$ on it, we characterize its irreducible $T$-invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank $2$ which are representable over $\mathbb{C}$.

math.CO↗

arXiv2021

Exploring tropical differential equations

The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-archimedean (e.g., $p$-adic) settings. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely) many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.

math.AG↗

arXiv2016

The Fundamental Theorem of Tropical Differential Algebraic Geometry

Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality $\text{trop}(V(I))=V(\text{trop}(I))$ between the tropicalization $\text{trop}(V(I))$ of the closed subscheme $V(I)\subset (K^*)^n$ and the tropical variety $V(\text{trop}(I))$ associated to the tropicalization of the ideal $\text{trop}(I)$. In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K[[t]]\{x_1,\ldots,x_n\}$, where $K$ is an uncountable algebraically closed field of characteristic zero. We define the tropicalization $\text{trop}(\text{Sol}(G))$ of the set of solutions $\text{Sol}(G)\subset K[[t]]^n$ of $G$, and the set of solutions associated to the tropicalization of the ideal $\text{trop}(G)$. These two sets are linked by a tropicalization morphism $\text{trop}:\text{Sol}(G)\longrightarrow \text{Sol}(\text{trop}(G))$. We show the equality $\text{trop}(\text{Sol}(G))=\text{Sol}(\text{trop}(G))$, answering a question raised by D. Grigoriev earlier this year.

math.AG↗