Midwest Algebraic Geometry Graduate Conference 2015

The Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago is pleased to host the Midwest Algebraic Geometry Graduate Conference for 2015.

The conference will feature nine talks beginning with the plenary lecture given at our departmental colloquium on Friday April 10th, followed by five talks on Saturday April 11th, and finishing with three talks on Sunday April 12th. The plenary talk will be given by Joe Harris of Harvard, and the talks on Saturday and Sunday will be given by graduate students in algebraic geometry.

#### Location and Dates:

The conference will take place at the University of Illinois, 851 South Morgan Street, Chicago, IL 60607 on April 10-12, 2015.

#### Deadline for Funding and to Submit an Abstract: February 15th, 2015.

##### Here is a list of registered particpants.(This list does not update immediately.)

Limited funding for lodging and travel is available for both participants and speakers. If you'd like to be considered to speak, please register with the above link and submit an abstract to uicmaggc@gmail.com.

Graduate organizers for this event are: Seckin Adali, Tim Ryan, Alexander Stathis, and Xudong Zheng. The faculty organizer is Izzet Coskun.

If you have any questions, then please contact us at uicmaggc@gmail.com.

Support provided by NSF RTG grant DMS-1246844.
Schedule

#### Rooms and times subject to change before the date of the event. Any updates will be announced via email.

##### Friday, April 10th, 2015
 Location Event 3:00 LC C6 Department Colloquium and Plenary LectureJoe Harris (Harvard) - TBA 4:15 SEO 300 Departmental Tea 5:30 DinnerWe will not provide dinner, but we will make reservations at a local restaurant for everyone who would like to attend.
##### Saturday, April 11th, 2015
 Location Event 9:00 SEO 300 BreakfastWe will provide coffee and assorted breakfast items. 9:30 LC C6 Clemens Koppensteiner - A Microlocal Characterization of Perverse Coherent Sheaves 10:45 LC C6 Eliana Duarte - Tensor Product Surfaces and Linear Syzygies 12:00 SEO 300 LunchProvided. 1:30 LC C6 Ed Dewey - Characteristic Classes of Cameral Covers 2:45 LC C6 David Bruce - Betti Tables of Graph Curves 4:00 LC C6 Robert Walker - Uniform Bounds Lurking in Affine Toric Rings
##### Sunday, April 12th, 2015
 Location Event 9:00 SEO 300 BreakfastWe will provide coffee and assorted breakfast items. 9:30 LC C6 Daniel Hast - The Geometry of "Short Intervals" in Function Field Arithmetic 10:45 LC C6 Jake Levinson - Limit Linear Series and Real Schubert Calculus 12:00 LC C6 Ashwath Rabindranath - Some Surfaces with Nonpolyhedral Nef Cones 1:15 SEO 300 LunchProvided.
Map

# The Talks

Joe Harris - Interpolation for Polynomials in Several VariablesFriday April 10th at 3:00pm
An elementary theorem says that we can always find a polynomial $f(x)$ of degree $d$ or less having specified values at $d+1$ given points $x$. When we try to state (let alone prove) an analogue for polynomials in several variables, however, we run into immediate difficulties. In this talk, I'll try to show that the difficulties lie in the geometry of the points, and suggest at least a conjectural answer to the problem.
Clemens Koppensteiner - A Microlocal Characterization of Perverse Coherent SheavesSaturday April 11th at 9:30am
The category of perverse sheaves on a complex variety is best understood via microlocal methods. Of particular interest is the compatibility with the vanishing cycles functor. It is natural to ask whether there is a similar viewpoint for perverse coherent sheaves on a scheme with a group action. I will give one answer to this question by taking local cohomology along "Lagrangian" subvarieties.
Eliana Duarte - Tensor Product Surfaces and Linear SyzygiesSaturday April 11th at 10:45am
A tensor product surface is the image of a map $\phi: \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3$. Such surfaces arise in geometric modeling, and it is often useful to find the implicit equation for the surface. In this talk, I will explain how the implicit equation can be obtained from the syzygies of the defining polynomials of the map via an approximation complex. In particular, the existence of a linear syzygy allows for a straightforward description of the implicit equation of the image from which we can describe part of the codimension 1 singular locus. This is joint work with Hal Schenk.
Ed Dewey - Characteristic Classes of Cameral CoversSaturday April 11th at 1:30pm
Suppose that you have a family of matrices parameterized by a scheme $X$. Taking Spec of the characteristic polynomial of the matrix associated to each point gives you a finite cover $X$, called the "spectral cover". The map taking a family of matrices to the spectral cover is called the Hitchin map.

Familes of operators generalize to certain gadgets called "Higgs Bundles" and spectral covers generalize to "cameral covers". Again there is a Hitchen map from one to the other, which has been useful for studying moduli spaces of principle bundles.

I will say a bit about the Hitchen map, and describe some work of Dima Arinkin and myself defining characteristic classes of cameral covers. The latter amounts to computing the cohomology ring of the moduli space of cameral covers and expressing its generators as Chern classes of certain natural vector bundles.

David Bruce - Betti Tables of Graph CurvesSaturday April 11th at 2:45pm
Given a graph one may obtain a reducible algebraic curve by associating a $\mathbb{P}^1$ to each vertex with two $\mathbb{P}^1$'s intersecting if there is an edge between the associated vertices. Such curves are called graph curves, or line arrangements, and were introduced by Bayer and Eisenbud to study Green's conjecture. I will discuss how the combinatorics of the graph affect the Betti table of its associated curve. In particular, I will present formulas for the Betti table for all graph curves of genus zero or one. Additionally, I will give formulas for the Betti numbers for a class of curves of higher genus. This talk is based on joint work with Pete Vermeire, Evan Nash, Ben Perez, and Pin-Hung Kao.
Robert Walker - Uniform Bounds Lurking in Affine Toric RingsSaturday April 11th at 4:00pm
I'll begin by motivating a statement of the Ein-Lazarsfeld-Smith (ELS) Theorem in its simplest form. The ELS theorem compares symbolic and ordinary powers of any ideal in any Noetherian regular ring $R$ containing a field. I'll mention at least one application or two of ELS to prove inclusions of ideals in "classical algebraic geometry" set ups where no other method is known. A similar result to ELS holds in smooth rings (coordinates rings of smooth affine varieties).

In structure, affine toric rings (coordinate rings of normal affine toric varieties) are "mused to be" the nicest class after regular rings on the one hand, and smooth rings on the other (since their singularities are rational). Starting from toric rings of dimension two or higher, I am searching for a result analogous to both ELS and the fact for smooth rings. The remainder of the talk will set up a statement of the conjecture I have in view, emphasizing work in progress to verify it in dimension two.

Daniel Hast - The Geometry of "Short Intervals" in Function Field ArithmeticSunday April 12th at 9:30am
Several major problems in analytic number theory, including the prime tuples conjecture and the Chowla conjecture, can be phrased as statements about the distribution of arithmetic functions in short intervals. We present a geometric approach to function field analogues of these problems, interpreting each moment of such a distribution as a point counting problem on a highly singular complete intersection. Computing geometric invarients of this variety (such as the dimension of the singular locus) leads to new asymptotic bounds on higher moments. This is joint work with Vlad Matei.
Jake Levinson - Limit Linear Series and Real Schubert CalculusSunday April 12th at 10:45am
We consider the moduli of linear series on marked stable curves, satisfying prescribed "vanishing conditions" with respect to each special point. These conditions arise naturally in the study of limit linear series: they describe compatibility constraints governing all linear series obtained by degenerating smooth curves into nodal curves.

For genus 0 curves, this moduli space (call it $S$) turns out to be very well-behaved: for instance, it is flat and Cohen-Macaulay over $\overline{M_{0,n}}$, and has the correct (expected) relative dimension. Moreover, in the relative-dimension-zero case, recent work due to Mukhin-Tarasov-Varchenko (2007) and Speyer (2014) showed that the real locus of $S$ is actually an unramified covering space of $\overline{M_{0,n}}(\mathbb{R})$; the monodromy of the cover has an elegant description in terms of objects from classical (combinatorial) Schubert calculus.

I'll discuss this construction, and some recent analogous work concerning the case where $S$ has relative dimension 1. In this case, there is interesting structure to be seen in both the real and complex geometry of the fibers.

Ashwath Rabindranath - Some Surfaces with Nonpolyhedral Nef ConesSunday April 12th at 12:00pm
The nef cone of a variety is an important invariant which captures important aspects of its birational geometry. We study the nef cones of certain surfaces and prove a new criterion for nef cones of surfaces to be nonpolyhedral. We apply this criterion to prove that the nef cone of $C \times C$ is not polyhedral when $C$ is a smooth projective curve of genus at least two over the complex numbers.
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