Browsing by Author "Hassett, Brendan"

Now showing 1 - 8 of 8
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    Effective Computation of Picard Groups and Brauer-Manin Obstructions of Degree Two K3 Surfaces Over Number Fields
    (Springer, 2013) Hassett, Brendan; Kresch, Andrew; Tschinkel, Yuri
    Using the Kuga-Satake correspondence we provide an effective algorithm for the computation of the Picard and Brauer groups of K3 surfaces of degree 2 over number fields.
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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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    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, Richard
    We 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.
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    Hodge Theory and Lagrangian Planes on Generalized Kummer Fourfolds
    (Independent University of Moscow, 2013) Hassett, Brendan; Tschinkel, Yuri
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    Infection By A String Link
    (2015-04-23) Vela, Diego; Hassett, Brendan; Harvey, Shelly; Cox, Steve; Leidy, Constance
    Satellite constructions on a knot can be thought of as taking some strands of a knot and then tying in another knot. Using satellite constructions one can construct many distinct isotopy classes of knots. Pushing this further one can construct distinct concordance classes of knots which preserve some algebraic invariants. Infection is a generalization of satellite operations which has been previously studied. An infection by a string link can be thought of as grabbing a knot at multiple locations and then tying in a link. Cochran, Friedl and Teichner showed that any algebraically slice knot is the result of infecting a slice knot by a string link(1). In this paper we use the infection construction to show that there exist knots which arise from infections by n-component string links that cannot be obtained by (n − 1)-component string links.
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    Log minimal model program for the moduli space of stable curves: the first flip
    (Department of Mathematics, Princeton University, 2013) Hassett, Brendan; Hyeon, Donghoon
    We give a geometric invariant theory (GIT) construction of the log canonical model M¯g(α) of the pairs (M¯g,αδ) for α∈(7/10–ϵ,7/10] for small ϵ∈Q+. We show that M¯g(7/10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; M¯g(7/10−ϵ) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction Ψ:M¯g(7/10+ϵ)→M¯g(7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction Ψ+:M¯g(7/10−ϵ)→M¯g(7/10) that is the Mori flip of Ψ.
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    Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence
    (De Gruyter, 2022) Frei, Sarah; Hassett, Brendan; Várilly-Alvarado, Anthony
    Given 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.
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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.