BAGELS 2023 Fall
BAGELS (Best Algebraic Geometry Eating and Learning Seminar) is a students-running algebraic geometry seminar. We meet weekly and share our knowledge and research on algebraic geometry and related areas. You can give a talk about what you are studying, reading, or researching. For past BAGELS talks, see BAGELS Spring 2023
During Fall 2023 semester, we meet on Thursdays from 3:30-4:30 in LCB 222.
Schedule
| Date | Speaker | Title/Abstract |
|---|---|---|
| Sep. 7 | Organizational Meeting | |
| Sep. 14 | Rahul Ajit | Birational Algebraic Geometry in Positive Characteristic and a Conjecture (I) Abstract: After recent breakthrough in the Classification of Varieties over complex number, it’s natural to try to understand the geometry of varieties defined over an algebraically closed field of char. p. Despite numerous difficulty, substantial progress has been made in recent works, and interesting Conjectures have been made. In my 1st talk, I’ll introduce the terminology required and mention some difficulty in positive characteristic. In my 2nd Talk, I’ll prove a result on klt (=strongly F-regular) singularity. |
| Sep. 21 | Rahul Ajit | Birational Algebraic Geometry in Positive Characteristic and a Conjecture (II) |
| Sep. 28 | Yu-Ting Huang | Correspondence between F-singularity and KLT/LC Singularity Abstract: Following Rahul’s talk, we will subtly investigate the relation between F-singularity over characteristic p fields and klt/lc singularity in characteristic 0 fields. Takagi proved that a pair being with klt singularity is equivalent to being locally strongly F-regular type. However, the relation between lc singularity and locally F-pure type has been conjectured for decades. So far, we only know this conjecture is equivalent to the Weak Ordinarity Conjecture due to Mustaţă, Srinivas, and Takagi. In this talk, I will first define all kinds of singularities we need. Then, I will prove the equivalence of the two conjectures. Lastly, I will discuss the progress and open problems related to this topic. |
| Oct. 5 | Shih-Hsin Wang | Families of Jets of Arc Type and Higher (co)Dimensional Du Val Singularities Abstract: We will reveal a natural correspondence between families of arcs originally studied by Nash and families of jets through the singular locus. Then, we focus on normal locally complete intersection varieties with rational singularities. In particular, we introduce the notion of higher Du Val singularities, a higher dimensional version of Du Val singularities that preserve many of their features, and more generally, the notion of higher compound Du Val singularities, whose definition parallels that of compound Du Val singularities. For such singularities, we prove a one-to-one correspondence exists between families of arcs and families of jets through the singularities. As an application, we give a solution to the Nash problem for higher Du Val singularities. |
| Oct. 19 | Yi-Heng Tsai | Graded Linear Series and Volume Functions Abstract: Given a line bundle L on a projective variety X, one can define the volume function of L as a limsup. It is well-known that if L is a big line bundle, the limsup is actually a limit. (One can find a proof in R. Lazarsfeld “Positivity in Algebraic Geometry” ) In this talk, following the approaches in “Newton-Okounkov Bodies, Semigroups of Integral Points, Graded Algebras and Intersection Theory”, written by Kiumars Kaven and A. G. Khovanskii, I will generalize the result to graded linear series on complete varieties. Moreover, the limit (i.e. the volume of a graded linear series) can be interpreted as a volume of the associated Newton-Okounkov body. To achieve the goal, I will establish a connection between graded algebras and semigroup of integral points, and then study the asymptotic behavior of the corresponding Hilbert functions. |
| Nov. 2 | Jorge Gaspar Lara | Descent and Weil Restriction of Scalars Abstract: I will introduce the notions of descent for modules and descent datum. I will particularly talk about flat descent, and I will sketch the proof of the correspondence between R-modules and S-modules with descent datum for a faithfully flat R-algebra S. After that I will discuss a representability criterion for functors that tells that if a functor is representable “locally”, then the functor is representable. Then we will use this result to give conditions for the Weil restriction of scalars of an affine group to be an affine group. Finally, if there is enough time, we will discuss flat descent for schemes and the notions of effectiveness and realization of descent datum. |
| Nov. 9 | Zach Mere | Fujita’s Vanishing Theorem Abstract: Recall the celebrated vanishing theorem due to Serre: given an ample line bundle L on a complex projective scheme, all the higher cohomology groups of an arbitrary coherent sheaf can be made to vanish by twisting by a sufficiently high power of L. A theorem of Fujita shows that Serre-type vanishings can be made to operate uniformly with respect to twists by nef divisors. In this talk, following Lazardsfeld’s Positivity in Algebraic Geometry, we’ll prove Fujita’s theorem and see several applications. |
| Nov. 16 | Jonathon Fleck | Castelnuovo-Mumford Regularity Abstract: Recall the Cartan-Serre-Grothendieck theorem which suggests that all of cohomological subtleties a coherent sheaf vanish when twisted by a sufficiently high multiple of the hyperplane line bundle. In this talk, we will introduce Castelnuovo-Mumford regularity and prove Mumford’s related theorem to show that this quantifies precisely when these subtleties vanish. We will then explore some properties, variants, and applications following Lazarsfeld’s Positivity. |
| Dec. 7 | Qingyuan Xue | Minimal Model Program for Foliations Abstract: In this talk, I will give a brief introduction on the Minimal Model Program for foliations, which can be viewed as a generalization of the Minimal Model Program for varieties. I will focus on some elementary results which are useful in the establishment of the Minimal Model Program. |