Algebraic Geometry Seminar
Department of Mathematics
Princeton University
Fall 2009 Lectures
Regular meeting time:
Tuesdays 4:30-5:30 (Tea served at 3:30)
Place: Fine 322
| Date | Speaker | Title |
| Sep 15 | Susan J. Sierra Princeton University |
Transversality and noncommutative geometry Birationally commutative graded algebras solve the moduli problem for "point modules" over a graded ring. They have been a fruitful source of counterexamples, examples, and intuition in noncommutative ring theory. We investigate when a large subclass of birationally commutative algebras is noetherian. Formally, these are idealizer subrings of twisted homogeneous coordinate rings. In the process, we give a (purely algebro-geometric) generalization of the Kleiman-Bertini theorem. |
| Sep 22 | No Seminar | |
| Sep 29 | Radu Laza Stony Brook University |
Notes on the birational geometry of moduli space of genus 4 curves In addition to the Deligne-Mumford compactification for the moduli space of genus four curves, there are a number of additional compactifications (such as those obtained by GIT and Kondo's ball quotient construction) that arise naturally. In this talk I will discuss joint work with Sebastian Casalaina-Martin where we compare some of these spaces. The description we obtain is similar to that for genus three curves (work of Hyeon-Lee), as well as to some previous results we have for the moduli space of cubic threefolds. |
| Oct 6 | Wenchuan Hu IAS |
Homotopy Theoretic methods on Chow varieties The homotopy theoretic method has been applied to the algebraic cycle theory for a long period of time. In particular, it can be applied to compute topological invariants of Chow varieties. In this talk I will discuss this method in calculating the Euler Characteristic of Chow varieties. The calculation in a direct and simple way (this result has been obtained by Blaine Lawson and Stephen Yau in a different way). This technique also can be applied to Chow varieties with certain group actions and other cases. Furthermore, I will also talk about the application of the method on l-adic Euler-Poincare Characteristic of Chow varieties over arbitrary algebraic closed field. |
| Oct 13 | JongHae Keum KIAS |
Q-homology projective planes and Montgomery-Yang Problem A Q-homology projective plane is a complex projective surface with quotient singularities only having the same Betti numbers as the complex projective plane. It is known that such surfaces can have at most 5 singular points. First, I will classify Q-homology projective planes with 5 quotient singular points, the maximum possible case. Next, some progress on the Algebraic Montgomery-Yang problem will be presented. Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Its algebraic version, as formulated by Kollár, predicts that every Q-homology projective plane has at most 3 singular points if its smooth locus is simply-connected |
| Oct 20 | Frans Oort University of Utrecht |
Algebraic curves with CM Is every abelian variety isogenous with the Jacobian of an algebraic curve? We will study also several other questions in arithmetic geometry and show various implications. We will mention some solutions to these problems. |
| Oct 27 | Mihnea Popa UIC |
BGG correspondence and the cohomology of compact Kaehler manifolds The cohomology algebra of the sheaf of holomorphic functions on a compact Kaehler manifold can be naturally viewed as a module over the exterior algebra of a vector space. A well-known result of Bernstein-Gel'fand-Gel'fand gives a correspondence between such "exterior" modules and linear complexes of modules over the symmetric algebra, i. e. the polynomial ring. I will explain how one can use a modern view on this correspondence, together with the Generic Vanishing theory developed by Green and Lazarsfeld via Hodge-theoretic methods, in order to understand subtle algebraic structures of the cohomology algebra. As a bonus, homological and commutative algebra tools can be applied on the polynomial ring side to obtain new inequalities for the holomorphic Euler characteristic and the Hodge numbers of compact Kaehler manifolds. This is joint work with R. Lazarsfeld. |
| Nov 3 | Fall Break | No Seminar |
| Nov 10 | Matt DeLand Stony Brook University |
Rational Simple Connectedness Abstract: Rational simple connectedness is an analog of simple connectedness for complex varieties having important applications: every 2-parameter family of rationally simply connected varieties has a rational section, and a 1-parameter family has so many rational sections that they approximate every power series section to arbitrary order. Unfortunately the condition is quite difficult to verify and is known to hold only for homogeneous spaces and also for some projective hypersurfaces satisfying a list of hypotheses. My new approach for verifying this condition works by studying a canonically defined foliation on the moduli space of rational curves on the variety. By proving integrability of this foliation, I prove every smooth complete intersections $X$ of type $(d_1, \ldots, d_c)$ in $\mathbb{P}^n$ is rationally simply connected whenever $\sum d_i^2 \leq n$ and when the associated moduli space of lines on $X$ is smooth. This degree range is sharp. |
| Nov 13 | Fine 214 3:00 pm:Vikraman Balaji Chennai Mathematical Institute and 4:30 pm:Tony Pantev University of Pennsylvania 6:00 pm:Dinner Nore special time and place |
Joint Columbia-Courant-Princeton Algebraic Geometry Seminar Vikraman Balaji: Analogue of the Narasimhan-Seshadri theorem in higher dimensions and holonomy We will discuss some recent work on natural analogues of the Narasimhan-Seshadri theorem on higher dimensional varieties with some applications to stable bundles on surfaces. The classical result related stability of bundles on projective smooth curves with irreducible unitary representations of the fundamental group. Analogues of holonomy groups and their representations play the corresponding role. Tony Pantev: Mirror symetry for del Pezzo surfaces I will discuss the general mirror symmetry question for del Pezzo surfaces in a setup that goes beyond the Hori-Vafa ansatz. I will describe the mirror map explicitly and will describe non-trivial consequences of homological mirror symmetry that can be proven directly. This is a joint work with Auroux, Katzarkov and Orlov. |
| Nov 17 | Paul Hacking University of Massachusetts Amherst |
Smoothing surface singularities via mirror symmetry
We use the Strominger-Yau-Zaslow interpretation of mirror symmetry to describe deformations of surface singularities in terms of counts of holomorphic curves and discs on a mirror surface. In particular we prove Looijenga's conjecture on smoothability of cusp singularities. This is joint work with Mark Gross and Sean Keel, and builds on work of Gross-Siebert and Gross-Pandharipande-Siebert. |
| Nov 24 | Tommaso deFernex University of Utah |
Rigidity properties of Fano varieties From the point of view of the Minimal Model Program, Fano varieties constitute the building blocks of uniruled varieties. Important information on the biregular and birational geometry of a Fano variety is encoded, via Mori theory, in certain combinatorial data corresponding to the Neron–Severi space of the variety. It turns out that, even when there is actual variation in moduli, much of such combinatorial data remains unaltered, provided that the singularities are "mild" in an appropriate sense. The talk is based on joint work with C. Hacon. |
| Dec 1 | Ekaterina Amerik IAS |
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| Dec 8 | Roya Beheshti Zavareh Washington University in St. Louis |
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Previous semester schedule (Spring
2009)
Other seminars in this department
For more information about this seminar, contact Guerogui Todorov (gtodorov@math).