The Yale Algebraic and Tropical Geometry seminar meets biweekly on Thursdays for two talks from 3:00 - 4:00 pm in Rosenfeld Hall (109 Grove Street) and then again from 4:15 - 5:15 in LOM 214.
Afterward, we usually go to dinner with the speaker at a local restaurant.
Ziwen Zhu, University of Utah, Equivariant K-stability and Valuative Criteria.
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It is conjectured that in order to test K-polystability of a
Fano variety, it is enough to consider only equivariant test
configurations with respect to a finite or connected reductive group
action. This has been proved for Fano manifolds and in singular cases,
only for torus actions. In this talk, I will talk about a valuative
criterion of equivariant K-stability with respect to an arbitrary group
action. This generalizes parallel results for usual K-stability.
October 24
Takumi Murayama, Princeton Univeristy, Singularities of generic projection hypersurfaces in arbitrary characteristic.
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Classically, it is known that every algebraic variety is birational to a hypersurface in some projective or affine space. Using generic linear projections, Doherty proved that over the complex numbers, this hypersurface can be taken to have at worst semi-log canonical singularities in dimensions up to five. This extends classically known results for curves and surfaces. We present a positive-characteristic analogue of Doherty's theorem, by showing that in the same dimensions, generic projection hypersurfaces in positive characteristic are F-pure. This proves cases of a conjecture of Bombieri, Andreotti, and Holm. To show our result, we study F-injective singularities, which are the analogue of Du Bois singularities in positive characteristic, and their behavior under flat morphisms. This work is joint with Rankeya Datta.
Man-Wai (Mandy) Cheung, Harvard University, Compactification for cluster varieties without frozen variables of finite type.
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Cluster varieties are blow up of toric varieties. They come in pairs (A,X), with A and X built from dual tori.
Compactifications of A, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the
compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the A and the X cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists
a positive polytope for the type A cluster varieties which give us a hint to the Batyrev-Borisov construction.
November 14
Benjamin Schroeter, State University of New York at Binghamton, Correlation Constants of Fields and Matroids.
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Given a finite connected graph and two of its edges, e and f. Choose a
spanning tree T uniformly at random. It follows from work of Kirchhoff
on electrical networks that the events e in T and f in T are negatively
correlated.
A combinatorial generalization of graphs and vector spaces is matroids.
I will discuss an analog of the above situation for general matroids,
thus introducing, as a measure of the correlation in a matroid, its
correlation constant. We use Hodge theory to bound these constants and
we give explicit constructions of realizable matroids with positively
correlated elements.
This is joint work with June Huh and Botong Wang.
Ralph Morrison, Williams College, The gonality of Cartesian products of graphs.
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Using the language of chip-firing, we can define a theory of divisors on graphs in parallel to the theory of divisors on algebraic curves. We can define the gonality of a graph, like the gonality of a curve, to be the minimum degree of a rank 1 divisor. It turns out that if we know the gonalities of two graphs, then we can find an upper bound on the gonality of the Cartesian product of the two graphs. I'll show many instances where this upper bound is the actual gonality, and also several constructions of graph products with gonality lower than expected. I'll also show that any nontrivial product graph satisfies Baker's gonality conjecture. This is joint work with Ivan Aidun.
December 5
Giulia Saccà, Columbia University, Birational geometry of the intermediate Jacobian fibration.
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A few year ago with Laza and Voisin we constructed a hyperkähler compactification of the intermediate Jacobian fibration associated to a general cubic fourfold. In this talk I will first show how such a compactification J(X) exists for any smooth cubic fourfold X and then discuss how the birational geometry of the fibration J(X) is governed by any extra algebraic cohomology classes on J(X).