Bagels talks - Fall 2019
|August 29th||Yen-An|| Multiplier Ideal Sheaf and its Application to Fujita's Conjecture
Abstract: Multiplier ideal sheaf plays an important role in algebraic geometry. For the first half of the talk, I will introduce the multiplier ideal sheaf and some of its properties. Then, for the second half, I will show how to make use of the multiplier ideal sheaf to get a quadratic bound for Fujita's base point freeness conjecture.
|September 5th||You-Cheng|| Intersection Theory
Abstract: In this talk, I will first recall the construction of normal cone and its properties. Then I will use it to define refined Gysin homomorphisms, which can be understood as the algebraic geometry version of cap product. As applications, I will talk about the ring structure on the Chow group of nonsingular variety and the construction of virtual fundamental class with a given moduli problem.
|September 12th||Seungsu|| Kawamata Viehweg vanishing theorem on globally F-regular varieties
Abstract: In general, there is a counter example for Kawamata Viehweg vanishing theorem on positive characteristic. However, we still have the vanishing theorem for the globally F-regular varieties. In this talk, we will briefly discuss the proof of Kodaira vanishing in characteristic 0 and then will show the Kawamata Viehweg vanishing on globally F regular varieties.
|September 19th||Marin||Birational geometry of moduli spaces
Abstract: In this talk, I will explain how to construct a nef line bundle on a moduli space of Bridgeland-semistable objects. This line bundle varies with the stability condition, which can change the moduli space birationally. As an application, one can use the stability conditions on the plane to run the minimal model program for the Hilbert scheme of points.
|September 26th||Christian||Geometric class field theory and generalizations
Abstract: Geometric class field theory constructs abelian covers of algebraic curves (with prescribed ramification). The key idea is to construct sheaves on the Picard group of the curve. In this talk we will discuss geometric class field theory, what it has to do with class field theory, and generalizations where the Picard group is replaced with the moduli space of rank n vector bundles.
|October 3rd||Matteo|| Bondal—Orlov’s Derived Reconstruction Theorem
Abstract: The derived category D(X) of a smooth projective variety X carries a great deal of the geometric information of X. It is a natural question to ask whether this information is enough to recover X uniquely — this is true, for example, in the case of the category of coherent sheaves Coh(X). In this talk I will briefly introduce D(X) and then prove Bondal and Orlov’s Theorem, which answers the previous questions positively in the case of K_X being ample (or anti-ample). I will also give a couple of counterexamples in the case of K_X=0 (the so-called Fourier-Mukai partners).
|October 10th||Fall Break||No talk|
|October 17th||Ziwen|| Volumes of Fano Manifolds
Abstract: The volume of a Fano manifold plays an important role in the study of Fano manifolds. One interesting question is to find upper bounds of the volumes for various classes of Fano manifolds. In this talk, I will consider Fano 3-folds first. By classification results of Fano 3-folds, we know that the volumes of Fano 3-folds are bounded by 64, which is the volume of the projective 3-space. I will also show how to compute explicitly upper bounds of the volumes for some particular classes of Fano 3-folds. This is not going to give any new upper bounds better than 64 for Fano 3-folds. However, similar questions become more interesting for higher dimensional Fano manifolds.
|October 24th||Qingyuan|| An Introduction to K-Stability
Abstract: Although K-stability is originally introduced by differential geometers, for the purpose of finding Kahler-Einstein metrics on Fano manifolds, it attracts more and more attention from algebraic geometers recently. One reason may be that K-stability seems to be a good condition when constructing the moduli space of Fano varieties. In this talk, I will first introduce the original algebraic definition of K-stability (by test configuration). Then I will introduce some useful invariants and state some theorems, including special test configuration, alpha invariants and delta invariants, valuative criterion, normalized volumes.
|October 31th||No talk
|November 7th||Hanlin||Etale cohomology and its application
Abstract: Etale cohomology is a powerful tool introduced by Grothendieck to prove Weil conjectures. The key idea is to define coverings for categories and construct etale sites on schemes. In this talk, we will see the construction of etale cohomology and some main results which are analogues of the ones in topology. Finally we will take a peek at Weil conjecture and how it relates to etale cohomology.
|November 14th||Lingyao||Non-Vanishing Theorem and Basepoint Free Theorem
Abstract: In order to run the MMP in higher dimensions, we need to develop the Cone Theorem for klt pairs. To prove the Cone Theorem we first use Non-Vanishing Theorem to prove Basepoint Free Theorem, combined with Rationality Theorem we can finally prove The Cone Theorem. I will prove the previous two in the seminar and will state the latter two if I have extra time. The proof will basically follow Kollar's book Birational Geometry on Algebraic Varieties.
|November 21th||Junpeng||Fujita Conjecture and effective base point freeness theorem
Abstract: For a smooth variety X and an ample divisor A on X, Fujita conjectured K_X+(n+2)A is very ample. The first breakthrough was achieved by Demailly who proved 2K_X+12n^nA is very ample. His method involves heavy analysis. In tomorrow's talk, I will introduce Kollar's work, he used algebraic method to find a universal bound m, such that K_X+mA is very ample.