arXiv subjects

Ángel David Ríos Ortiz

Publications and source records attributed to Ángel David Ríos Ortiz.

13 recordsLinked to original sources

arXiv2026

Projective models for Hilbert squares of $K3$ surfaces

For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ associated to quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for $g\geq 7$. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for $S$. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly similar fashion. In consequence, we give explicit descriptions of its ideal and syzygies. Furthermore, we extract new information on the locally complete families, in a first step towards the understanding of their projective geometry.

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arXiv2026

The SYZ conjecture for singular moduli spaces of sheaves on K3 surfaces

In this paper we prove the SYZ conjecture and we compute the Huybrechts--Riemann--Roch polynomial for irreducible symplectic varieties that are locally trivial deformation equivalent to moduli spaces of sheaves on K3 surfaces. As an intermediate step in the argument, we generalise to the singular setting a result of Matsushita about moduli spaces of lagrangian fibrations of primitive symplectic varieties.

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arXiv2026

Bogomolov-Guan Manifolds are not formal

Bogomolov--Guan manifolds form the only known series of compact simply connected non-Kähler holomorphic symplectic manifolds. We prove that all Bogomolov--Guan manifolds of complex dimension at least four are nonformal over $\mathbb{Q}$. The proof constructs a uniform degree-eight $a$-Massey product in a finite-dimensional invariant de Rham model of the symmetric quotient of the Bogomolov--Guan precursor. Its indeterminacy vanishes, and its nontriviality is detected by an explicit top-degree pairing arising from an $S_{n+1}$-invariant tensor contraction. A nonzero-degree map then transfers nonformality to the smooth Bogomolov--Guan manifold. We also compare the Bogomolov--Guan precursor with the nilmanifold appearing in Guan's original construction, showing that they are related by a natural $S_{n+1}$-equivariant finite cover of degree $(n+1)^2$ and have the same invariant de Rham model. Using Guan's resolution, which we prove to be projective and semismall, together with Saito's decomposition theorem, we further show that the third Betti number of every Bogomolov--Guan manifold of complex dimension at least four is equal to $3$.

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arXiv2026

Coble type hypersurfaces and hyperkähler fourfolds

A classical result, already observed by Coble, asserts that a genus 2 Jacobian can be embedded in $\mathbf P^8$ as the singular locus of a unique cubic hypersurface; similarly, the Kummer of a genus 3 Jacobian is embedded in $\mathbf P^7$ as the singular locus of a unique quartic hypersurface. We present a precise analogue of these results in the context of hyperkähler fourfolds: for the general member in the 20-dimensional locally complete families of polarized hyperkähler fourfolds of $\mathrm{K3}^{[2]}$-type with squares 4 or 6 and divisibility 1, we establish the existence of a unique Coble type hypersurface. As a consequence, we describe several geometric aspects of the corresponding moduli spaces.

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arXiv2026

MMP for Enriques pairs and singular Enriques varieties

We introduce and study the class of primitive Enriques varieties, whose smooth members are Enriques manifolds. We provide several examples and we demonstrate that this class is stable under the operations of the Minimal Model Program (MMP). In particular, given an Enriques manifold $Y$ and an effective $\mathbb{R}$-divisor $B_Y$ on $Y$ such that the pair $(Y,B_Y)$ is log canonical, we prove that any $(K_Y+B_Y)$-MMP terminates with a minimal model $(Y',B_{Y'})$ of $(Y,B_Y)$, where $Y'$ is a $\mathbb{Q}$-factorial primitive Enriques variety with canonical singularities. Finally, we investigate the asymptotic theory of Enriques manifolds.

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arXiv2025

Hilbert schemes of points on fold-like curves and their combinatorics

We investigate the Hilbert scheme of points on curves with n-fold singularities, that is curves that look locally around their singular points as the axis in an affine space. We describe the structure and number of its irreducible components, and provide a detailed analysis of their singularities, revealing rich combinatorial patterns governing its geometry.

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arXiv2025

Coble duality for Jacobian Kummer fourfolds

We study projective models of generalized Kummer fourfolds via O'Grady's theta groups and the classical Coble cubic. More precisely, we establish a duality between two singular models of the generalized Kummer fourfold of a Jacobian abelian surface. We also give projective models for singular Jacobian Kummer varieties of arbitrary dimension. Along the way, we also construct a first non-natural involution on the Hilbert square of a Jacobian surface. In the appendix, we study singularities of secants of arbitrary varieties at identifiable points, following Choi, Lacini, Park and Sheridan.

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arXiv2024

Decomposing Tensor Spaces via Path Signatures

The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor.

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arXiv2024

Asymptotic base loci on hyper-Kähler manifolds

Given a projective hyper-Kähler manifold $X$, we study the asymptotic base loci of big divisors on $X$. We provide a numerical characterization of these loci and study how they vary while moving a big divisor class in the big cone, using the divisorial Zariski decomposition, and the Beauville-Bogomolov-Fujiki form. We determine the dual of the cones of $k$-ample divisors $\mathrm{Amp}_k(X)$, for any $1\leq k \leq \mathrm{dim}(X)$, answering affirmatively (in the case of projective hyper-Kähler manifolds) a question asked by Sam Payne. We provide a decomposition for the effective cone $\mathrm{Eff}(X)$ into chambers of Mori-type, analogous to that for Mori dream spaces into Mori chambers. To conclude, we illustrate our results with several examples.

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arXiv2024

Crossing the transcendental divide: from Schottky groups to algebraic curves

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles in the complex plane via free groups of Möbius transformations called Schottky groups. We construct a family of non-hyperelliptic surfaces of genus $g\geq 3$ where we know the Riemann surface as well as properties of the canonical embedding, including a nontrivial symmetry group and a real structure with the maximal number of connected components (an $M$-curve). We then numerically approximate the algebraic curve and Riemann matrices underlying our family of Riemann surfaces.

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arXiv2023

Riemann-Roch Polynomials of the known Hyperkähler Manifolds

We compute explicit formulas for the Euler characteristic of line bundles in the two exceptional examples of Hyperkähler Manifolds introduced by O'Grady. In the Appendix Yalong Cao and Chen Jiang use our formulas to compute the Chern numbers for the 10-dimensional example.

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arXiv2023

On the transcendental lattices of Hyperkähler manifolds

We introduce the notion of a Hyper-Kähler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-Kähler manifolds studying those that are induced by a K3 or abelian surface, giving lattice-theoretic criteria to decide whether or not they are birational to a moduli space of sheaves over said surface. We highlight the different behaviors we find for the particular class of hyper-Kähler manifolds of O'Grady type.

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arXiv2023

A survey on rational curves on complex surfaces

In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the Kähler and non-Kähler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight the different approaches applied to the study of rational curves in each class. We also provide several examples and point out some open problems.

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