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Patricio Gallardo

Publications and source records attributed to Patricio Gallardo.

At least 19 recordsLinked to original sources
arXiv2026

Geography and Deformations of $\mathbb{Z}_2^s$-Covers of General Type Over Weighted Projective Threefolds

We study threefolds of general type constructed as $\mathbb{Z}_2^s$-covers of weighted projective spaces with a particular focus on their invariants, deformation theory, and the behavior of the $m$-canonical map. For the invariants, we write the ratios of the volume to the topological and holomorphic Euler characteristics as functions of the ratios of the degree of the branch divisors with respect to the total degree. From this expression, we obtain their asymptotic behavior, bounds, and a counterexample to a conjecture made by Bruce Hunt about the non-existence of smooth threefolds in a forbidden zone. From the perspective of deformation theory, we extend the criterion for such covers to be general in their moduli to the case when the weighted projective threefold has isolated singularities and the cover is non-flat, i.e., the pushforward of the structure sheaf splits as a direct sum of reflexive sheaves as opposed to line bundles. As an application, we present new numerical criteria for constructing components of the moduli spaces of stable threefolds and give concrete examples illustrating their application. Finally, we introduce techniques from Fourier transforms on finite groups to completely classify when a $\mathbb{Z}_2^s$-cover is a flat pluricanonical map. For $s \geq 2$, there are $32$ deformation types. We also show that there exist non-flat canonical and bicanonical $\mathbb{Z}_2^s$-covers for arbitrarily large values of $s$.

math.AG↗
arXiv2026

Understanding Tool-Augmented Agents for Lean Formalization: A Factorial Analysis

Automatic translation of natural language mathematics into faithful Lean 4 code is hindered by the fundamental dissonance between informal set-theoretic intuition and strict formal type theory. This gap often causes LLMs to hallucinate non-existent library definitions, resulting in code that fails to compile or lacks semantic fidelity. In this work, we investigate the effectiveness of tool-augmented agents for this task through a systematic factorial analysis of three distinct tool categories: Fine-tuned Model Querying (accessing expert drafts), Knowledge Search (retrieving symbol definitions), and Compiler Feedback (verifying code via a Lean REPL). We first benchmark the agent against one-shot baselines, demonstrating large gains in both compilation success and semantic equivalence. We then use the factorial decomposition to quantify the impact of each category, isolating the marginal contribution of each tool type to overall performance.

cs.SE↗
arXiv2024

Polymatroids and moduli of points in flags

We introduce and study different compactifications of the moduli space of $n$ distinct weighted labeled points in a flag of affine spaces. We construct these spaces via the weighted and generalized Fulton-MacPherson compactifications of Routis and Kim-Sato. For certain weights, our compactifications are toric and isomorphic to the polypermutohedral and polystellahedral varieties, which arise in the work of Crowley-Huh-Larson-Simpson-Wang and Eur-Larson on polymatroids, a generalization of matroids. Moreover, we show that these toric compactifications have a fibration structure, with fibers isomorphic to the Losev-Manin space, and are related to each other via a geometric quotient.

math.AG↗
arXiv2024

An explicit wall crossing for the moduli space of hyperplane arrangements

The moduli space of hyperplanes in projective space has a family of geometric and modular compactifications that parametrize stable hyperplane arrangements with respect to a weight vector. Among these, there is a toric compactification that generalizes the Losev-Manin moduli space of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. As an application of our work, we show that any $\mathbb{Q}$-factorialization of the blow up at the identity of the torus of the generalized Losev-Manin space is not a Mori dream space for a sufficiently high number of hyperplanes. Additionally, for lines in the plane, we provide a precise description of the wall crossing.

math.AG↗
arXiv2023

Eigenperiods and the moduli of points in the line

We study the period map of configurations of n points on the projective line constructed via a cyclic cover branching along these points. By considering the decomposition of its Hodge structure into eigenspaces, we establish the codimension of the locus where the eigenperiod map is still pure. Furthermore, we show that the period map extends to the divisors of a specific moduli space of weighted stable rational curves, and that this extension satisfies a local Torelli map along its fibers.

math.AG↗
arXiv2023

Variation of stability for moduli spaces of unordered points in the plane

We study compactifications of the moduli space of unordered points in the plane via variation of GIT quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.

math.AG↗
arXiv2023

Enumeration of max-pooling responses with generalized permutohedra

We investigate the combinatorics of max-pooling layers, which are functions that downsample input arrays by taking the maximum over shifted windows of input coordinates, and which are commonly used in convolutional neural networks. We obtain results on the number of linearity regions of these functions by equivalently counting the number of vertices of certain Minkowski sums of simplices. We characterize the faces of such polytopes and obtain generating functions and closed formulas for the number of vertices and facets in a 1D max-pooling layer depending on the size of the pooling windows and stride, and for the number of vertices in a special case of 2D max-pooling.

math.CO↗
arXiv2023

Progress in the Design of the Atacama Large Aperture Submillimeter Telescope

The Atacama Large Aperture Submillimeter Telescope (AtLAST) aims to be the premier next generation large diameter (50 meter) single dish observatory capable of observations across the millimeter/submillimeter spectrum, from 30~GHz to 1~THz. AtLAST will be sited in Chile at approximately 5100 meters above sea level, high in the Atacama Desert near Llano de Chajnantor. The novel rocking-chair telescope design allows for a unprecedentedly wide field of view (FoV) of 1-2$^\circ$ diameter, a large receiver cabin housing six major instruments, and high structural stability during fast scanning operations (up to $\sim 3^\circ$ per second in azimuth). Here we describe the current status of, and expected outcomes for, the antenna design study, which will be completed in 2024.

astro-ph.IM↗
arXiv2023

Computation of GIT quotients of semisimple groups

We describe three algorithms to determine the stable, semistable, and torus-polystable loci of the GIT quotient of a projective variety by a reductive group. The algorithms are efficient when the group is semisimple. By using an implementation of our algorithms for simple groups, we provide several applications to the moduli theory of algebraic varieties, including the K-moduli of algebraic varieties, the moduli of algebraic curves and the Mukai models of the moduli space of curves for low genus. We also discuss a number of potential improvements and some natural open problems arising from this work.

math.AG↗
arXiv2023

Higher-dimensional Losev-Manin spaces and their geometry

The classical Losev-Manin space is a toric compactification of the moduli space of $n$ points in the affine line modulo translation and scaling. Motivated by this, we study its higher-dimensional toric counterparts, which compactify the moduli space of $n$ distinct labeled points in affine space modulo translation and scaling. We show that these moduli spaces are a fibration over a product of projective spaces -- with fibers isomorphic to the Losev-Manin space -- and that they are isomorphic to the normalization of a Chow quotient. Moreover, we present a criterion to decide whether the blow-up of a toric variety along the closure of a subtorus is a Mori dream space. As an application, we demonstrate that a related generalization of the moduli space of pointed rational curves constructed by Chen, Gibney, and Krashen is not a Mori dream space when the number of points is at least nine, regardless of the dimension.

math.AG↗
arXiv2023

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

Smooth minimal surfaces of general type with $K^2=1$, $p_g=2$, and $q=0$ constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space $\mathbf{M}$ of their canonical models admits a modular compactification $\overline{\mathbf{M}}$ via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of $\mathbf{M}$ and the Hodge theory of the degenerate surfaces that the eight divisors parametrize.

math.AG↗
arXiv2022

The Atacama Cosmology Telescope: The Persistence of Neutrino Self-Interaction in Cosmological Measurements

We use data from the Atacama Cosmology Telescope (ACT) DR4 to search for the presence of neutrino self-interaction in the cosmic microwave background. Consistent with prior works, the posterior distributions we find are bimodal, with one mode consistent with $Λ$CDM and one where neutrinos strongly self-interact. By combining ACT data with large-scale information from WMAP, we find that a delayed onset of neutrino free streaming caused by significantly strong neutrino self-interaction is compatible with these data at the $2-3σ$ level. As seen in the past, the preference shifts to $Λ$CDM with the inclusion of Planck data. We determine that the preference for strong neutrino self-interaction is largely driven by angular scales corresponding to $700 \lesssim \ell \lesssim 1000$ in the ACT E-mode polarization data. This region is expected to be key to discriminate between neutrino self-interacting modes and will soon be probed with more sensitive data.

astro-ph.CO↗
arXiv2021

Quivers and moduli of their thin sincere representations in Macaulay2

We introduce the Macaulay2 package ThinSincereQuivers for studying acyclic quivers, the moduli of their thin-sincere representations, and the reflexive flow polytopes associated to them. We provide some background on the topic and illustrate how the package recovers examples from the literature.

math.AG↗
arXiv2021

CCAT-prime: Designs and status of the first light 280 GHz MKID array and Mod-Cam receiver

The CCAT-prime project's first light array will be deployed in Mod-Cam, a single-module testbed and first light cryostat, on the Fred Young Submillimeter Telescope (FYST) in Chile's high Atacama desert in late 2022. FYST is a six-meter aperture telescope being built on Cerro Chajnantor at an elevation of 5600 meters to observe at millimeter and submillimeter wavelengths.1 Mod-Cam will pave the way for Prime-Cam, the primary first generation instrument, which will house up to seven instrument modules to simultaneously observe the sky and study a diverse set of science goals from monitoring protostars to probing distant galaxy clusters and characterizing the cosmic microwave background (CMB). At least one feedhorn-coupled array of microwave kinetic inductance detectors (MKIDs) centered on 280 GHz will be included in Mod-Cam at first light, with additional instrument modules to be deployed along with Prime-Cam in stages. The first 280 GHz detector array was fabricated by the Quantum Sensors Group at NIST in Boulder, CO and includes 3,456 polarization-sensitive MKIDs. Current mechanical designs allow for up to three hexagonal arrays to be placed in each single instrument module. We present details on this first light detector array, including mechanical designs and cold readout plans, as well as introducing Mod-Cam as both a testbed and predecessor to Prime-Cam.

astro-ph.IM↗
arXiv2021

The cross correlation of the ABS and ACT maps

One of the most important checks for systematic errors in CMB studies is the cross correlation of maps made by independent experiments. In this paper we report on the cross correlation between maps from the Atacama B-mode Search (ABS) and Atacama Cosmology Telescope (ACT) experiments in both temperature and polarization. These completely different measurements have a clear correlation with each other and with the Planck satellite in both the EE and TE spectra at $\ell<400$ over the roughly $1100$ deg$^2$ common to all three. The TB, EB, and BB cross spectra are consistent with noise. Exploiting such cross-correlations will be important for future experiments operating in Chile that aim to probe the $30<\ell<8,000$ range.

astro-ph.CO↗