# Graduate Algebraic Geometry Seminar Spring 2018

When: Wednesdays 3:40pm

Where:Van Vleck B321 (Spring 2018)

Lizzie the OFFICIAL mascot of GAGS!!

Who: All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.

Why: The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.

How:If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is here.

## Give a talk!

We need volunteers to give talks this semester. If you're interested contact Juliette or Moisés, or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.

## Being an audience member

The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:

• Do Not Speak For/Over the Speaker:

## Wish List

Here are the topics we're DYING to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.

### Specifically Vague Topics

• D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.
• Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)

### Interesting Papers & Books

• Symplectic structure of the moduli space of sheaves on an abelian or K3 surface - Shigeru Mukai.
• Residues and Duality - Robin Hatshorne.
• Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)
• Coherent sheaves on P^n and problems in linear algebra - A. A. Beilinson.
• In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)
• Frobenius splitting and cohomology vanishing for Schubert varieties - V.B. Mehta and A. Ramanathan.
• In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off!
• Schubert Calculus - S. L. Kleiman and Dan Laksov.
• An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!
• Rational Isogenies of Prime Degree - Barry Mazur.
• In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.
• Esquisse d’une programme - Alexander Grothendieck.
• Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)
• Géométrie algébraique et géométrie analytique - J.P. Serre.
• A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)
• Limit linear series: Basic theory- David Eisenbud and Joe Harris.
• One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.
• Picard Groups of Moduli Problems - David Mumford.
• This paper is essentially the origin of algebraic stacks.
• The Structure of Algebraic Threefolds: An Introduction to Mori's Program - Janos Kollar
• This paper is an introduction to Mori's famous minimal model program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties.
• Cayley-Bacharach Formulas - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.
• A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: What is equation for this cubic curve?.
• On Varieties of Minimal Degree (A Centennial Approach) - David Eisenbud and Joe Harris.
• Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.
• The Gromov-Witten potential associated to a TCFT - Kevin J. Costello.
• This seems incredibly interesting, but fairing warning this paper has been described as highly technical, which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)

## Spring 2017

 Date Speaker Title (click to see abstract) February 14 Moisés Herradón Cueto Fun with commutative groups February 21 Moisés Herradón Cueto $\displaystyle{ \mathcal F }$un with commutative groups February 28 Dima Arinkin ODEs: algebraic vs analytic vs formal March 7 Vladimir Sotirov TBD March 14 David Wagner TBD March 21 TBD TBD March 28 Spring break Whoo! April 4 Rachel Davis TBD April 11 Brandon Boggess TBD April 18 Soumya Sankar TBD April 25 Solly Parenti TBD May 2 TBD TBD

## February 14

 Moisés Herradón Cueto Title: Fun with commutative groups Abstract: My goal is to next week talk about Gerard Laumon's preprint Transformation de Fourier généralisée, which will encapsulate all the Fourier transforms I can name (which for me includes the Fourier transform for D-modules on affine space, the Mellin transform, the Fourier-Mukai transform for abelian varieties and the rare John Mahoney transform between modules on the punctured line and vector spaces with a Z-action) into one crazy package. In order to achieve this altered mental state, we will have to rethink all our preconceptions, and rediscover algebraic groups, make sense of Cartier duality for them, define formal groups along the way, and see how duality of Abelian varieties relates to Cartier duality. If you come to the talk in an already in this altered mental state, then hopefully I'll do some examples that you've been too lazy to work out for yourself.

## February 21

 $\displaystyle{ \mathcal F }$un with commutative groups Title: $\displaystyle{ \mathcal F }$un with commutative groups Abstract: I will talk about how last week's Cartier duality relates to duality for abelian varieties. Then we will see how this allows us to take certain complexes of groups which are self dual, going back to altered mental states which are now derived. If we're going to Fourier transform something, it should be a sheaf, but what is even a sheaf on a complex of groups? Once we see what it should be, I will handwave what the Fourier transform is and give a bunch of concrete examples.

## February 28

 Dima Arinkin Title: ODEs: algebraic vs analytic vs formal Abstract: Linear ordinary differential equations (or, as they are known geometrically, bundles with connections on Riemann surfaces) can be studied from many directions. In algebraic geometry, the goal is to classify equations rather than to solve them. The classification can be done in different settings: algebraic, analytic, or formal; each setting has its advantages and disadvantages. However, for one of the most important class of equations (equations with regular singularities aka Fuchsian equations) the three approaches agree, leading to a rich and beautiful picture. In my talk, I will sketch the algebraic theory of differential equations, focusing on equations with regular singularities. (I do not expect to have time for irregular singularities.) The talk is related to my topics course on D-modules, but it does not rely on the topics course.

## March 7

 TBD Title: TBD Abstract: TBD

## March 14

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## March 21

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## April 4

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## April 11

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## April 18

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## April 25

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## May 2

 TBD Title: TBD Abstract: TBD