Spring 2019

Spring 2019 Algebraic Geometry near-Boston Conference

When: April 27, 2019, 10am-5pm

Where: Room 2-449, Mathematics Department, MIT

Speakers:

Registration: To help out the organizers with the headcount, please fill out a short registration form.

Schedule:

10-11 Igor Krylov: Parameter spaces of del Pezzo fibrations and birational geometry

11-11:20 Coffee break

11:20-12:20 Zhiyuan Li: Arithmetic period map and Shafarevich type question for HK varieties

12:20-2 Lunch break

2-3 Brooke Ullery: The gonality of complete intersection curves

3-3:20 Coffee break

3:20-4:20 Yunqing Tang: Reductions of abelian surfaces over global function fields

Titles and abstracts:

Igor Krylov: Parameter spaces of del Pezzo fibrations and birational geometry

Del Pezzo fibrations are one of the types of the Mori Fiber Space output of the MMP. There may be many models for the del Pezzo fibration and we would like to work with the best one. For example it is known that for conic bundles there exists a model with a smooth total space. I will describe a construction of parameter space of del Pezzo surfaces of degree 1 and 2. Using this parameter space I define what are the best models of del Pezzo fibrations of degrees 1 and 2. Then I show the existence of a good birational model.

Zhiyuan Li: Arithmetic period map and Shafarevich type question for HK varieties

Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the question for the finiteness of the set of isomorphism classes of varieties over K and with good reduction outside of S. Faltings and Andre have proved this conjecture for curves of fixed genus, polarized abelian surfaces and HK varieties. In this talk, I will talk about the unpolarized Shafarevich type questions and explain how it relates to other finiteness results for hyperkahler varieties. This is an ongoing project with Lie Fu.

Yunqing Tang: Reductions of abelian surfaces over global function fields

For a non-isotrivial ordinary abelian surface $A$ over a global function field with everywhere good reduction, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the product of two elliptic curves. This result can be viewed as a generalization of a theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.

Brooke Ullery: The gonality of complete intersection curves

The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. If a curve is embedded in projective space, it is natural to ask whether the gonality is related to the embedding. In my talk, I will discuss work with James Hotchkiss. Our main result is that, under mild degree hypotheses, the gonality of a complete intersection curve in projective space is computed by projection from a codimension 2 linear space, and any minimal degree branched covering of P^1 arises in this way.