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Full-Text Articles in Algebraic Geometry
Spherical Tropicalization, Anastasios Vogiannou
Spherical Tropicalization, Anastasios Vogiannou
Doctoral Dissertations
In this thesis, I extend tropicalization of subvarieties of algebraic tori over a trivially valued algebraically closed field to subvarieties of spherical homogeneous spaces. I show the existence of tropical compactifications in a general setting. Given a tropical compactification of a closed subvariety of a spherical homogeneous space, I show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the closed subvariety. I provide examples of tropicalization of subvarieties of GL(n), SL(n), and PGL(n).
Skein Theory And Algebraic Geometry For The Two-Variable Kauffman Invariant Of Links, Thomas Shelly
Skein Theory And Algebraic Geometry For The Two-Variable Kauffman Invariant Of Links, Thomas Shelly
Doctoral Dissertations
We conjecture a relationship between the Hilbert schemes of points on a singular plane curve and the Kauffman invariant of the link associated to the singularity. Specifcally, we conjecture that the generating function of certain weighted Euler characteristics of the Hilbert schemes is given by a normalized specialization of the difference between the Kauffman and HOMFLY polynomials of the link. We prove the conjecture for torus knots. We also develop some skein theory for computing the Kauffman polynomial of links associated to singular points on plane curves.
Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, Nikolay Buskin
Algebraicity Of Rational Hodge Isometries Of K3 Surfaces, Nikolay Buskin
Doctoral Dissertations
Consider any rational Hodge isometry $\psi:H^2(S_1,\QQ)\rightarrow H^2(S_2,\QQ)$ between any two K\"ahler $K3$ surfaces $S_1$ and $S_2$. We prove that the cohomology class of $\psi$ in $H^{2,2}(S_1\times S_2)$ is a polynomial in Chern classes of coherent analytic sheaves over $S_1 \times S_2$. Consequently, the cohomology class of $\psi$ is algebraic whenever $S_1$ and $S_2$ are algebraic.
Topology Of The Affine Springer Fiber In Type A, Tobias Wilson
Topology Of The Affine Springer Fiber In Type A, Tobias Wilson
Doctoral Dissertations
We develop algorithms for describing elements of the affine Springer fiber in type A for certain 2 g(C[[t]]). For these , which are equivalued, integral, and regular, it is known that the affine Springer fiber, X, has a paving by affines resulting from the intersection of Schubert cells with X. Our description of the elements of Xallow us to understand these affine spaces and write down explicit dimension formulae. We also explore some closure relations between the affine spaces and begin to describe the moment map for the both the regular and extended torus action.