arXiv subjects

Ignacio Barros

Publications and source records attributed to Ignacio Barros.

15 recordsLinked to original sources

arXiv2026

NEXP-Completeness and Exponential Coefficient Growth for Existential Presburger Arithmetic with Divisibility

We prove that satisfiability for existential Presburger arithmetic with divisibility (EPAD) is NEXP-hard. Together with the known NEXP upper bound, this establishes NEXP-completeness. The lower bound is obtained by encoding succinct Boolean formulas whose satisfying assignments may have exponential length. A central difficulty is to represent, within a polynomial-size EPAD formula, the exponentially long integers arising in this encoding. To address this difficulty, we introduce fixed-width arithmetic logic circuits (ALCs), whose gates perform addition, multiplication, and bit shifts. We show that polynomial-size uniform ALC families compute exactly the functions in FPSPACE, while their succinct exponential-size counterpart computes exactly the functions in FEXP. In both cases, the computed functions admit polynomial-size functional definitions in EPAD. This provides the compressed arithmetic needed for the lower-bound reduction. The same construction yields NEXP-hardness for nonerasing word equations with Presburger length constraints over a fixed two-letter alphabet. We also study the elimination of divisibility constraints by enumerating their possible quotients. Earlier work identified an NP fragment in which the variables can be ordered so that divisibility dependencies always move forward through the order. We introduce a different fragment, called merge-absorptive, in which bounded-quotient divisibilities can successively connect all variable components. This fragment is polynomial-time recognizable and admits complete finite-quotient elimination, yet its satisfiability problem remains NEXP-complete. Finally, we show that the elimination process may necessarily produce equations with exponentially large coefficients.

cs.LO↗

arXiv2026

Finite generation of Noether-Lefschetz divisors and the slope of the moduli space of cubic fourfolds

We study divisors on moduli spaces of cubic fourfolds with simple singularities and of quasi-polarized K3 surfaces of degree $2d$. For the moduli space of cubic fourfolds, we introduce a slope quantity to characterize the effective cone and prove an explicit bound for it. For the K3 moduli spaces, we give an explicit finite presentation of the rational Picard group by showing that it is generated by Noether-Lefschetz divisors of discriminant less than or equal to $4d$. As a byproduct, we obtain two explicit expressions for the Hodge class in terms of Noether-Lefschetz divisors, and we indicate analogous results for higher-codimension Noether-Lefschetz cycles.

math.AG↗

arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗

arXiv2025

Extremal divisors on moduli spaces of K3 surfaces

We establish criteria for when Noether--Lefschetz divisors generate an extremal ray in the cone of pseudoeffective divisors of an orthogonal modular variety. In particular, we exhibit many extremal rays of the cone of pseudoeffective divisors on any moduli space~$\mathcal{F}_{2d}$ of quasi-polarized K3 surfaces of degree $d$, as well as on any normal projective $\mathbb{Q}$-factorial compactification $\overline{\mathcal{F}}_{2d}$ of $\mathcal{F}_{2d}$ lying over the Baily--Borel compactification.

math.AG↗

arXiv2025

A simple criterion for the uniruledness of an orthogonal modular variety

We exhibit a simple uniruledness criterion for general orthogonal modular varieties in terms of invariants of the corresponding lattice. As an application, we obtain the uniruledness of almost all Nikulin--Vinberg moduli spaces parameterizing projective K3 surfaces of Picard number at least 3 and fixed finite automorphism group.

math.AG↗

arXiv2024

Cones of Noether-Lefschetz divisors and moduli spaces of hyperk\"ahler manifolds

We give a general formula for generators of the NL-cone, the cone of effective linear combinations of irreducible components of Noether-Lefschetz divisors, on an orthogonal modular variety. We then fully describe the NL-cone and its extremal rays in the cases of moduli spaces of polarized K3 surfaces and hyperk\"ahler manifolds of known deformation type for low degree polarizations. Moreover, we exhibit explicit divisors in the boundary of NL-cones for polarizations of arbitrarily large degrees. Additionally, we study the NL-positivity of the canonical class for these modular varieties. As a consequence, we obtain uniruledness results for moduli spaces of primitively polarized hyperk\"ahler manifolds of ${\rm{OG6}}$ and ${\rm{Kum}}_n$-type. Finally, we show that any family of polarized hyperk\"ahler fourfolds of ${\rm{Kum}}_2$-type with polarization of degree $2$ and divisibility $2$ over a projective base is isotrivial.

math.AG↗

arXiv2023

On the irrationality of moduli spaces of projective hyperk\"ahler manifolds

The aim of this paper is to estimate the irrationality of moduli spaces of hyperk\"ahler manifolds of types K3$^{[n]}$, Kum$_{n}$, OG6, and OG10. We prove that the degrees of irrationality of these moduli spaces are bounded from above by a universal polynomial in the dimension and degree of the manifolds they parametrize. We also give a polynomial bound for the degrees of irrationality of moduli spaces of $(1,d)$-polarized abelian surfaces.

math.AG↗

arXiv2022

Kodaira dimension of moduli spaces of hyperk\"ahler varieties

We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahler varieties deformation equivalent to the Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional variety. This question was studied by Gritsenko-Hulek-Sankaran in the cases of $K3^{[2]}$ and OG10 type when the divisibility of the polarization is one. We generalize their results to higher dimension and divisibility. As a main result, for almost all dimensions $2n$ we provide a lower bound on the degree such that for all higher degrees, every component of the moduli space of polarized hyperk\"ahler varieties of $K3^{[n]}$ type is of general type.

math.AG↗

arXiv2021

The Kodaira classification of the moduli of hyperelliptic curves

We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces have non $\mathbb{Q}$-factorial singularities. We complete the Kodaira classification by proving that these spaces have Kodaira dimension $4g+3$ when the number of markings is $4g+6$ and are of general type when the number of markings is $n\geq4g+7$. Similarly, we consider the natural finite cover given by ordering the Weierstrass points. In this case, we provide a full Kodaira classification showing that the Kodaira dimension is negative when $n\leq3$, one when $n=4$, and of general type when $n\geq 5$. For this, we carry out a singularity analysis of ordered and unordered pointed Hurwitz spaces. We show that the ordered space has canonical singularities and the unordered space has non-canonical singularities. We describe all non-canonical points and show that pluricanonical forms defined on the full regular locus extend to any resolution. Further, we provide a full classification of the structure of the pseudo-effective cone of Cartier divisors for the moduli space of hyperelliptic curves with marked points. We show the cone is non-polyhedral when the number of markings is at least two and polyhedral in the remaining cases.

math.AG↗

arXiv2020

On the irrationality of moduli spaces of K3 surfaces

We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon>0$ and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree $10$. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperk\"ahler fourfolds.

math.AG↗

arXiv2020

Pencils on surfaces with normal crossings and the Kodaira dimension of $\overline{\mathcal{M}}_{g,n}$

We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline{\mathcal{M}}_{g,n}$ is not pseudo-effective in some range, implying that $\overline{\mathcal{M}}_{12,6},\overline{\mathcal{M}}_{12,7},\overline{\mathcal{M}}_{13,4}$ and $\overline{\mathcal{M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline{\mathcal{M}}_{12,8}$ and $\overline{\mathcal{M}}_{16}$. We also show that the moduli of $(4g+5)$-pointed hyperelliptic curves $\mathcal{H}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the Kodaira classification for moduli of pointed hyperelliptic curves.

math.AG↗

arXiv2020

Two moduli spaces of Calabi-Yau type

We show $\overline{\mathcal{M}}_{10,10}$ and $\mathcal{F}_{11,9}$ have Kodaira dimension zero. Our method relies on the construction of a number of curves via nodal Lefschetz pencils on blown-up $K3$ surfaces. The construction further yields that any effective divisor in $\overline{\mathcal{M}}_{g}$ with slope $<6+(12-δ)/(g+1)$ must contain the locus of curves that are the normalization of a $δ$-nodal curve lying on a $K3$ surface of genus $g+δ$.

math.AG↗

arXiv2019

On product identities and the Chow rings of holomorphic symplectic varieties

For a moduli space $M$ of stable sheaves over a $K3$ surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\star (M \times X^\ell),\, \ell \geq 1,$ generalizing the classic Beauville-Voisin identity for a $K3$ surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_\star (M) \subset CH_\star (M).$ The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on $CH_\star (M)$, which we also discuss. We prove the proposed identities when $M$ is the Hilbert scheme of points on a $K3$ surface.

math.AG↗

arXiv2018

Geometry of the moduli space of $n$-pointed K3 surfaces of genus 11

We prove that the moduli space of polarized $K3$ surfaces of genus eleven with $n$ marked points is unirational when $n\leq 6$ and uniruled when $n\leq7$. As a consequence, we settle a long standing but not proved assertion about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked points has non-negative Kodaira dimension.

math.AG↗

arXiv2018

Uniruledness of Strata of Holomorphic Differentials in Small Genus

We address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus $3\leq g\leq6$, we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most $g-1$, hence showing in addition that these strata are unirational.

math.AG↗