University of Utah
Algebraic Geometry Seminar
Fall 2009: Tuesdays 3:30-4:30, LCB 225

Schedule of upcoming talks:

Click on the title of a talk for the abstract (if available).

Date Speaker Title
September 8 Roi Docampo Álvarez
Arcs on determinantal varieties
September 15
Martin Deraux
(Université de Grenoble I)
Holomorphic maps between ball quotients A recent theorem of Koziarz and Mok shows that there can be no holomorphic submersions between compact ball quotients, apart from totally geodesic coverings. After sketching the proof of their result, I will describe the known examples of holomorphic surjections between ball quotients, namely the Livne fibrations and the Mostow-Toledo maps, that illustrate the failure of superrigidity of the automorphism group of complex hyperbolic space. These two classes of examples can be understood in terms of "forgetful maps" between Deligne-Mostow ball quotients, which allows to extend the list of examples a little.
September 22
Amin Gholampour
BPS invariants for resolutions of polyhedral singularities Let G be a finite subgroup of SO(3). Y=G-Hilb(C^3) gives the preferred Calabi-Yau resolution of singularities of C^3/G. We use two different approaches to determine the BPS invariants of Y: Gromov-Witten theory, and a specific moduli space of torsion sheaves on Y. The result is expressed in terms of an ADE root system canonically associated to G.
September 29
Jim Carlson
(Clay Mathematics Institute)
Speculations on cubic surfaces: Hodge and Galois theory I will discuss work in progress with Domingo Toledo. (a) What can one say about the locus of cubic surfaces with periods certain number fields? (Periods to defined: not what you might think they are). (b) What can one say about their Galois representations? The main object of study is a certain natural l-adic representation whose image we hope to prove is large.
October 6 Yunfeng Jiang
Counting invariants for O+O(-2)-quiver representations
October 20 Davide Fusi
Rationality and related problems
October 24-25 WAGS / Fall 2009
October 27
Luca Scala
(University of Chicago)
Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles We will motivate the work by an important example of the Strange Duality conjecture for moduli spaces of sheaves on the projective plane, in close relation with Barth morphism. The technique used to attack the conjecture in this case, due to Le Potier, He and Danila, uses moduli spaces of coherent systems in order to interpolate the moduli space M_n (of semistable sheaves of rank 2, degree 0 and second Chern class n) with a Hilbert scheme of points in P_2; this procedure allows to reduce the computation of the space of global sections of the determinant line bundle on M_n to the understanding of the cohomology H^*(P_2^[m] , S^q L^[m] ) of the Hilbert scheme of points on P_2 with values in symmetric powers of a tautological bundle L^[m], associated to a certain line bundle L on P_2 . Danila's results on the cohomology of S^q L_[m] yield the strange duality for 0 < n < 19. We will show how to improve and generalize Danila's results on the cohomology of the Hilbert scheme of points on a smooth surface X with values in representations of tautological bundles. By making use of the derived McKay correspondence of Bridgeland-King-Reid, adapted by Haiman in the case of the Hilbert scheme X^[m], we prove general formulas for the cohomology of X^[m] with values in the double tensor power and general exterior powers of tautological bundles. Finally, we will sketch the work in progress on higher symmetric powers of tautological bundles, directly useful for the strange duality conjecture on the projective plane. (arxiv: 0710.3072)
November 3
Sam Payne
(Stanford University)
Topology of compactified tropicalizations I will introduce compact tropicalizations of subvarieties of toric varieties over nonarchimedean fields, and discuss how the topology of these compactified tropicalizations relate to analytifications and weight filtrations.
November 10
Christian Liedtke
(Stanford University / Universität Düsseldorf)
Elliptic K3 surfaces with p^n-torsion sections We classify K3 surfaces in positive characteristic p, which come with an elliptic fibration and a section of order p^n. Since such elliptic fibrations pull back from the universal elliptic curve over the Igusa curve, which is the characteristic p-analog of the modular curves, we get explicit equations. In particular, we can check arithmetic conjectures and find beautiful connections between arithmetic and geometry.
November 17
David Treumann
(Northwestern University)
The coherent-constructible correspondence for toric varieties This is a talk on joint work with Bohan Fang, Chiu-Chu Melissa Liu, and Eric Zaslow. An equivariant ample line bundle on a toric variety determines a polytope in a real vector space. I will discuss work we have done to extend this familiar correspondence to an equivalence of categories, between the derived category of coherent sheaves on the toric variety and a derived category of polyhedrally-constructible sheaves on the real vector space. Many famous aspects of the dictionary between toric geometry and polyhedral combinatorics are visible in this equivalence. Work of Nadler and Zaslow relates the second category to a Fukaya category of Lagrangian submanifolds of a symplectic vector space. Thus our correspondence can be viewed as a verification--for toric varieties--of a version of Kontsevich's homological mirror symmetry conjecture.
November 24 No seminar today
December 1 Jimmy Dillies
Invariants of Dessins
December 8
Marco Aldi
(University of California, Berkeley)
Categorical Representations and Mirror Symmetry Following ideas of Kontsevich-Soibelman, Tamarkin and Kajiura we study certain categorical representations of the (affine) symplectic group. Our main application is to a conjecture of Ben-Bassat regarding homological mirror symmetry for torus bundles over tori.

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