Browsing by Author "Várilly-Alvarado, Anthony"
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Item Abelian n -Division Fields of Elliptic Curves and Brauer Groups of Product Kummer & Abelian Surfaces(Cambridge University Press, 2017) Várilly-Alvarado, Anthony; Viray, BiancaLet Y be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of Q. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes BrY/Br1Y is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic 0, we prove that the existence of a strong uniform bound on the size of the odd torsion of BrY/Br1Y is equivalent to the existence of a strong uniform bound on integers n for which there exist non-CM elliptic curves with abelian n-division fields. Using the same methods we show that, for a fixed prime ℓ, a number field k of fixed degree r, and a fixed discriminant of the geometric Néron–Severi lattice, #(BrY/Br1Y)[ℓ∞] is bounded by a constant that depends only on ℓ, r, and the discriminant.Item Brauer groups of a family of nonnegative Kodaira dimension elliptic surfaces(2023-04-12) Zheng, Ken; Várilly-Alvarado, AnthonyWe explore the Brauer groups of the elliptic surfaces given by y² = x³ + t⁶ᵐ + 1 over Q for m = 2, 3. When m = 2, the resulting surface is K3, and when m = 3, the surface is honestly elliptic with Kodaira dimension 1. We compute the algebraic Brauer groups of these surfaces by studying the action of Gal(̅Q/Q) on their Neron-Severi groups. Following the work of Gvirtz, Loughran, and Nakahara, we find bounds for the exponents of transcendental Brauer groups of these surfaces. The transcendental Brauer group is closely related to the transcendental lattice. The argument begins with an explicit description of the basis of the respective transcendental lattices and reinterpreting elements of these lattices as elements in rings of integers. From this, we bound the transcendental Brauer group. These bounds apply more generally to the surfaces given by y² = x³ + A₁t⁶ᵐ + A₂ for integers Aᵢ and m = 2, 3.Item Brauer groups of Kummer surfaces arising from elliptic curves with complex multiplication(2019-04-19) Johnson, Alexis Katherine; Várilly-Alvarado, AnthonyThe Brauer group of a variety often captures arithmetic information about the space. In this thesis, we study the Brauer group of a special kind of K3 surface, namely, a Kummer surface associated with a self-product of an elliptic curve over a number field with complex multiplication by a non-maximal order in an imaginary quadratic number field. Skorobogatov has conjectured that the Brauer group controls the existence of rational points on K3 surfaces. In practice, given a K3 surface, one often needs explicit descriptions of Brauer elements in order to study the behavior of rational points. The surfaces studied in this thesis have a geometrically rich structure that enables us to explicitly compute both the algebraic and transcendental Brauer groups over the rational numbers. Furthermore, over arbitrary number fields, we bound the transcendental Brauer group.Item Campana points of bounded height on vector group compactifications(Wiley, 2021) Pieropan, Marta; Smeets, Arne; Tanimoto, Sho; Várilly-Alvarado, AnthonyWe initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behavior of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our setup, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups.Item Cohomology classes responsible for Brauer-Manin obstructions, with applications to rational and K3 surfaces(2018-04-18) Nakahara, Masahiro; Várilly-Alvarado, AnthonyWe study the classes in the Brauer group of varieties that never obstruct the Hasse principle. We prove that for a variety with a genus 1 fibration, if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer–Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties. We also analyze the Brauer–Manin obstruction to rational points on the K3 surfaces over Q given by double covers of P^2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer–Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.Item Cubic fourfolds containing a plane and a quantic del Pezzo surface(Foundation Compositio Mathematica, 2014) Auel, Asher; Bernardara, Marcello; Bolognesi, Michele; Várilly-Alvarado, AnthonyWe isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C8 ∩ C14 has five irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaffian and has β nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov’s derived categorical conjecture on the rationality of cubic fourfolds.Item Explicit computation of symmetric differentials and its application to quasihyperbolicity(Mathematical Science Publishers, 2022) Bruin, Nils; Thomas, Jordan; Várilly-Alvarado, AnthonyWe develop explicit techniques to investigate algebraic quasihyperbolicity of singular surfaces through the constraints imposed by symmetric differentials. We apply these methods to prove that rational curves on Barth’s sextic surface, apart from some well-known ones, must pass through at least four singularities, and that genus 1 curves must pass through at least two. On the surface classifying perfect cuboids, our methods show that rational curves, again apart from some well-known ones, must pass through at least seven singularities, and that genus 1 curves must pass through at least two. We also improve lower bounds on the dimension of the space of symmetric differentials on surfaces with A1-singularities, and use our work to show that Barth’s decic, Sarti’s surface, and the surface parametrizing 3×3 magic squares of squares are all algebraically quasihyperbolic.Item Geometric Invariant Theory Quotient of the Hilbert Scheme of Six Points on the Projective Plane(2015-06-29) Durgin, Natalie Jean; Hassett, Brendan; Várilly-Alvarado, Anthony; Grandy, RichardWe provide an asymptotic stability portrait for the Hilbert scheme of six points on the complex projective plane, and provide a description of its geometric invariant theory (GIT) quotient.Item Kodaira dimensions of some moduli spaces of special hyperkähler fourfolds(2020-08-10) Petok, Jack; Várilly-Alvarado, Anthony; Goldman, RonWe study the Noether-Lefschetz locus of the moduli space $\mathcal{M}$ of $K3^{[2]}$-fourfolds with a polarization of degree $2$. Following Hassett's work on cubic fourfolds, Debarre, Iliev, and Manivel have shown that the Noether-Lefschetz locus in $\mathcal{M}$ is a countable union of special divisors $\mathcal{M}_d$, where the discriminant $d$ is a positive integer congruent to $0,2,$ or $4$ modulo 8. In this thesis, we compute the Kodaira dimensions of these special divisors for all but finitely many discriminants; in particular, we show that for $d>176$ and for many other small values of $d$, the space $\mathcal{M}_d$ is a variety of general type. The main idea of the proof is to study the Kodaira dimension of the moduli spaces using the "quasi-pullback" trick of Gritsenko-Hulek-Sankaran: by explicitly constructing certain modular forms on the period domain, we can show the plurigenera of a smooth compactification of $\mathcal{M}_d$ grow fast enough to conclude that $\mathcal{M}_d$ is of general type for all but $40$ values of $d$. We also give information about the Kodaira dimension of $\mathcal{M}_d$ for 6 additional values of $d$, leaving only 34 values of $d$ for which we cannot yet say anything about the Kodaira dimension.Item Level structures on Abelian varieties, Kodaira dimensions, and Lang's conjecture(Elsevier, 2018) Abramovich, Dan; Várilly-Alvarado, AnthonyAssuming Lang's conjecture, we prove that for a prime p, number field K, and positive integer g, there is an integer r such that no principally polarized abelian variety A/K has full level-pr structure. To this end, we use a result of Zuo to prove that for each closed subvariety X in the moduli space Ag of principally polarized abelian varieties of dimension g, there exists a level mX such that the irreducible components of the preimage of X in Ag[m] are of general type for m>mX.Item Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence(De Gruyter, 2022) Frei, Sarah; Hassett, Brendan; Várilly-Alvarado, AnthonyGiven a smooth projective variety over a number field and an element of its Brauer group, we consider the specialization of the Brauer class at a place of good reduction for the variety and the class. We are interested in the case of K3 surfaces. We show that a Brauer class on a very general polarized K3 surface over a number field becomes trivial after specialization at a set of places of positive natural density. We deduce that there exist cubic fourfolds over number fields that are conjecturally irrational, with rational reduction at a positive proportion of places. We also deduce that there are twisted derived equivalent K3 surfaces which become derived equivalent after reduction at a positive proportion of places.Item Symbolic solution for computational quantum many-body theory development(2018-03-02) Zhao, Jinmo; Scuseria, Gustavo E; Wolynes, Peter G; Várilly-Alvarado, AnthonyComputational many-body theories in quantum chemistry, condensed matter, and nuclear physics aim to provide sufficiently accurate description of and insights into the motion of many interacting particles. Due to their intrinsic complexity, the development of such theories generally involves very complex, tedious, and error-prone symbolic manipulations. Here a complete solution to automate the symbolics in many-body theory development is attempted. General data structures based on an existing computer algebra system are designed to specifically address the symbolic problems for which there is currently no satisfactory handling. Based on the data structures, algorithms are given to accomplish common symbolic manipulations and simplifications. Noncommutative algebraic systems, tensors with symmetry, and symbolic summations can all enjoy deep simplifications efficient enough for theories of very complex form. After the symbolic derivation, novel algorithms for automatic optimization of tensor contractions and their sums are devised, which can be used together with automatic code generation tools. In this way, the burden of symbolic tasks in theory development can be vastly reduced, with the potential to spare scientists more time and energy for the actual art and science of many-body theories.Item The inverse Galois problem for del Pezzo surfaces of degree 1 and algebraic K3 surfaces(2022-08-12) Wolff, Stephen Heinz; Várilly-Alvarado, AnthonyIn this thesis we study the inverse Galois problem for algebraic K3 surfaces and for del Pezzo surfaces of degree one. We begin with an overview of how the question of the existence of k-points on a nice k-variety leads, via Brauer groups, to the inverse Galois problem. We then discuss an algorithm to compute all finite subgroups of the general linear group GL(n,Z), up to conjugacy. The first cohomology of these subgroups are a superset of the target groups of the inverse Galois problem for any family of nice k-varieties whose geometric Picard group is free and of finite rank. We apply these results to algebraic K3 surfaces defined over the rational numbers, providing explicit equations for a surface solving the only nontrivial instance of the inverse Galois problem in geometric Picard rank two. We then study the inverse Galois problem for representatives from three families of del Pezzo surfaces of degree one, searching for 5-torsion in the Brauer group. For two of the three surfaces, we show that the Brauer group is trivial when the surface is defined over the rational numbers, but becomes isomorphic to Z/5Z or (Z/5Z)^2 when the base field is raised to a suitable number field. For the third surface, we show that its splitting field has degree 2400 as an extension of the rational numbers, a degree consistent with 5-torsion in the Brauer group.