Browsing by Author "Hassett, Brendan E."

Now showing 1 - 7 of 7
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    Deformations of Hilbert Schemes of Points on K3 Surfaces and Representation Theory
    (2014-04-22) Zhang, Letao; Hassett, Brendan E.; Hardt, Robert M.; Riviere, Beatrice M.
    We study the cohomology rings of Kaehler deformations X of Hilbert schemes of points on K3 surfaces by representation theory. We compute the graded character formula of the Mumford - Tate group representation on the cohomology ring of X. Furthermore, we also study the Hodge structure of X, and find the generating series for deducting the number of canaonical Hodge classes in the middle cohomology.
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    Density of rational points on K3 surfaces over function fields
    (2012-09-05) Li, Zhiyuan; Hassett, Brendan E.; Wolf, Michael; Rojo, Javier
    In this paper, we study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and the subgroup of the N´eron-Severi group generated by these sections is not finitely generated. This also gives examples of K3 surfaces over the function field F of a complex curve with Zariski dense F-rational points, whose geometric models are Calabi-Yau. Furthermore, we also generalize our results to the cases of families of higher dimensional Calabi-Yau varieties with Calabi-Yau ambient spaces.
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    Geometric compactification of moduli space of cubic surfaces and Kirwan blowup
    (2005) Zhang, Jun; Hassett, Brendan E.
    We compactify the moduli space of cubic surfaces by constructing a moduli stack of stable cubic and triploid surfaces, together with a projective coarse moduli space. We also give a detailed geometric interpretation. This compactification coincides with the Naruki Cross Ratio Variety. We also show that this compactification can be obtained by taking a certain weighted blowup along the strictly semi-stable stratum from Kirwan's stratification and partial desingularization procedures.
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    On the smooth linear section of the Grassmannian Gr(2, n)
    (2012) Xu, Fei; Hassett, Brendan E.
    In this thesis, we will study the smooth linear section of the Grassmannian Gr(2, n). Explicitly, we give a criterion for the rationality of such linear section in terms of its codimension in the Plü ̈cker embedding in projective space. Moreover, to obtain a better understanding of the birational parametrization of these linear sections, we analyze their Hodge structures in the cases of even and odd codimensions. To be more precise, we provide numerous examples which suggest certain patterns of Hodge diamonds corresponding to even and odd cases and derive the proof of general patterns for codimension 3 smooth linear section of Gr(2, n) corresponding to odd and even n.
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    Rational points on del Pezzo surfaces of degree 1 and 2
    (2011) Li, Shuijing; Hassett, Brendan E.
    One of the fundamental problems in Algebraic Geometry is to study solutions to certain systems of polynomial equations in several variables, or in other words, find rational points on a given variety which is defined by equations. In this paper, we discuss the existence of del Pezzo surface of degree 1 and 2 with a unique rational point over any finite field [Special characters omitted.] , and we will give a lower bound on the number of rational points to each q. Furthermore, we will give explicit equations of del Pezzo surfaces with a unique rational point. Also we will discuss the rationality property of the del Pezzo surfaces especially in lower degrees.
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    Weak approximation for degree 2del Pezzo surfaces at places of bad reduction
    (2007) Knecht, Amanda Leigh; Hassett, Brendan E.
    This thesis addresses weak approximation for certain degree 2 del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of singular reductions of the surfaces to find prescribed sections through these fibers.
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    Young tableaux with applications to representation theory and flag manifolds
    (2010) Bruun, Christian; Hassett, Brendan E.
    We outline the use of Young tableaux to describe geometric and algebraic objects using combinatorial methods. In particular, we discuss applications to representations of the symmetric group and the general linear group, flag varieties, and Schubert varieties. We also describe some recent work, including proofs of the Saturation Conjecture and a theorem on the eigenvalues of sums of Hermitian matrices.