Browsing by Author "Varilly-Alvarado, Anthony"

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    A Bayesian approach to computing Brauer groups of cubic surfaces
    (2022-12-02) James, Austen A; Varilly-Alvarado, Anthony
    We present an algorithm for computing the Brauer groups of cubic surfaces. The algorithm takes as input an equation for a cubic surface X and a confidence threshold 0.5 < r < 1 and outputs a candidate for the Brauer group of X and a confidence level > r for the result. The algorithm runs by sampling lifts of Frobenius at many primes of good reduction and relies on Chebotarev’s density theorem and Bayesian inference to produce, with confidence level > r, a subgroup of the Weyl group of E_6. This subgroup represents the action of Galois on the geometric Picard group of X, from which we compute the Brauer group of X. We give a description of this algorithm and a proof that it terminates, as well as an implementation in Magma. We also examine the speed of such an approach relative to existing methods and explore how the Bayesian technique of this algorithm can be applied to answer questions concerning the Galois and Brauer groups of other classes of surfaces.
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    Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups
    (Elsevier, 2014) Varilly-Alvarado, Anthony; Viray, Bianca
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    Failure of the Hasse Principle on General K3 Surfaces
    (Oxford University Press, 2013) Hassett, Brendan; Varilly-Alvarado, Anthony
    We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X, ) is constructed from a double cover of P2 × P2 ramified over a hypersurface of bi-degree (2, 2).
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    From Infinite to Finite: Rational Reductions of del Pezzo Surfaces
    (2024-08-08) Spaulding, Zac James; Varilly-Alvarado, Anthony
    It is well-known that all del Pezzo surfaces of degree at least 5 over a finite field are rational, i.e., birational to the projective plane, but this is generally not true for those of lower degree. If we fix a del Pezzo surface X of degree d < 5, defined over a number field k, and consider the primes p of k of good reduction for X, then we may ask: how often do we expect X_p, the reduction of X modulo p, to be rational? To answer this question, we combine a result of Colliot-Thélène from 2019 with the Chebotarev Density Theorem to determine the natural density of the set \pi_{rat}(X,k) --- the set of primes of k for which the reduction X_p is F_p-rational --- in the set of all primes of k. We present an algorithm to determine this natural density with input data being the action of the absolute Galois group of k on the geometric Picard group. We implement this algorithm in magma, exhibiting the nonzero uniform lower bound 1/1920 for this natural density, independent of starting data.
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    Notes on Real Rationally Connected Varieties and Fano Threefolds of Genus 12
    (2016-11-01) Allums, Derek; Hassett, Brendan E; Varilly-Alvarado, Anthony
    We show that a smooth projective geometrically rationally connected variety over the real numbers with at least one rational point admits a non-constant mapping from a smooth projective curve. Additionally, we show that many real smooth Fano complete intersections admit non-constant maps from the real anisotropic conic. Furthermore, we compute the genus and degree of the singular locus of the locus of lines on a genus 12 Fano threefold. After blowing up this locus to obtain simple normal crossings divisor, we compute the cohomology of the complement, in which we see the genus of this curve appear in weight 5 of the third cohomology group.
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    Toric fibrations and models of universal torsors
    (2015-04-23) Kozin, Nikita; Hassett, Brendan; Varilly-Alvarado, Anthony; Dobelman, John
    We study smooth projective threefolds fibered by toric surfaces over the projective line. We show that for certain families of degree 6 del Pezzo and quadric surface bundles the universal torsor corresponding to the generic fiber extends to a smooth model over the base. It respects the action of model for the Neron-Severi torus and induces the Abel-Jacobi map from the space of sections. This corresponds to the map from a set of rational points on the generic fiber to the Galois cohomology group of torsors under the Neron-Severi torus. For the model of the latter we also compute corresponding groups of torsors over the base.