About 

This is a one day algebraic geometry event, happening on Saturday, March 25, 2023 at the University of Michigan. 

Schedule 

All talks will be held in 1360 East Hall. Refreshments will be available in the atrium. 

9:30-10:00 Refreshments 

10:00-10:50 Lena Ji 

11:10-12:00 Nick Rekuski 

12:00-2:00 Lunch break

2:00-3:00 Jacob Lurie

3:00-3:30 Refreshments 

3:30-4:00 Yizhen Zhao

4:30-5:00 James Hotchkiss

6:00- Conference dinner 

Speakers and abstracts

• James Hotchkiss (University of Michigan)

The period-index problem over the complex numbers

The period-index problem is a longstanding question about the complexity of Brauer classes over a field. I will discuss some Hodge-theoretic aspects of the problem for complex function fields, and give some applications to Brauer groups and the integral Hodge conjecture.

• Lena Ji (University of Michigan)

Rationality of conic bundle threefolds over non-closed fields

Over the complex numbers, rationality of conic bundles over P^2 is well understood: it is characterized by the Clemens–Griffiths intermediate Jacobian obstruction to rationality. In this talk, we investigate the rationality of these conic bundles over non-closed fields. We study the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg, which extends the classical obstruction over C. We focus on the case when the discriminant curve has degree 4, which is the first case where geometric rationality and existence of a k-rational point do not characterize k-rationality. This talk is based on joint work with S. Frei–S. Sankar–B. Viray–I. Vogt and joint work with M. Ji.

• Jacob Lurie (Institute for Advanced Study)

A Prismatic Perspective on Deligne-Illusie

Let X be a projective algebraic variety over the field of complex numbers. A central result of Hodge theory is that every cohomology class on X can be represented (uniquely) by a harmonic differential form. This has an algebraic consequence: the Hodge-to-de-Rham spectral sequence of X is degenerate. In the late 1980's, Deligne and Illusie gave a purely algebraic proof of the latter statement, using positive characteristic algebraic geometry in place of classical Hodge theory. In this talk, I'll explain a new perspective on the work of Deligne-Illusie, using ideas from the theory of prismatic theory (joint work with Bhargav Bhatt).

• Nick Rekuski (Wayne State University)

Stability of Kernel Bundles

It is notoriously difficult to construct stable bundles with given Chern classes. This difficultly is partially because the relationship between categorical constructions and stability is unclear. For example, it is unknown whether there exists conditions that guarantee the kernel of a morphism between stable bundles is still stable. In this direction, we present some results showing when a kernel sheaf—which is the kernel of the evaluation map on global sections—associated to a globally generated, slope stable sheaf is also slope stable.

• Yizhen Zhao (Michigan State University)

Symbol length of p-torsion Brauer groups and Wild Ramifications  

Let F be a field of characteristic p and A be an F-central simple algebra of period-p. It is known that any such A is Brauer equivalent to the tensor product of symbol algebras of index-p. The symbol length of A is the minimal number of symbol algebras in such an expression. We use Kato's Swan conductor to give a new method to find the symbol length of period-p Brauer classes over complete discrete valuation fields with residue fields with p-basis of rank 1. The quotient fields of completions of the local rings of a surface at codimension 1 points are examples of such fields. Then we can use these results to analyze wild ramification along codimension 1 points as Brauer groups have purity in codimension 1. If time permits, we will discuss an application to show there is no tamely ramified Brauer class over a curve over \bar{F_p}(t) with good reduction. We may also discuss current work in progress in which we are trying to approach the p-symbol length problem of the function fields of curves over \bar{F_p}((t)) using the main theorem. 

Registration 

If you would like to attend, please fill out the registration form, where you can indicate whether you would like to attend the conference dinner or be considered for travel funding. 

The registration deadline is March 3, 2023 for those who wish to attend the conference dinner, and March 17, 2023 otherwise. 

Contact

Please send any questions to arper@umich.edu.