Boston Algebraic Geometry Day 

Speakers: Nathan Chen, Richard Haburcak,  Eric Larson, Anda Tenie 

Location: Tufts University Joyce Cummings Center (JCC) room 502, 177 College Ave, Medford, MA 02155

Public Transportation:  The green line E has its final stop next to the JCC building. The Davis square stop of the red line is a 15 minute walk.

Parking:  There is metered parking on College Avenue close to the JCC building. Parking is free and plentiful on College Avenue and parallel streets beyond George Street (see the map above). Alternatively, you can purchase a Tufts parking permit and park at the large parking lot across the street from JCC.  

Schedule: 

9:30-10  Check-in and refreshments
10-11 Nathan Chen: Exploring variaties through their birational automorphism groups
11-11:30  Coffee break
11:30-12:30  Richard Haburcak: Refined Brill--Noether theory
12:30-2:30  Lunch (not provided)
2:30-3:30 Anda Tenie: Reider-Type Theorems on Normal Surfaces via Bridgeland Stability
3:30-4  Coffee break
4-5  Eric Larson: Normal bundles of rational curves in Grassmannians

Abstracts: 

Nathan Chen (Harvard University): Exploring variaties through their birational automorphism groups

In this talk, we will study how the birational automorphism group can sometimes be used to shed light on the geometry of a variety. This is joint work with a number of collaborators, including L. Ji, D. Stapleton, and separately L. Esser, A. Regeta, C. Urech, I. van Santen.

Richard Haburcak (Ohio State University): Refined Brill--Noether theory

Classical Brill--Noether theory studies linear systems on a general genus g curve. However, most curves we encounter are not general. A refined Brill--Noether theory studies the linear systems on curves with a given Brill--Noether special linear system, which can be rephrased as understanding the relative positions of (components of) Brill--Noether loci, which parameterize curves with a particular linear series. We'll survey classical and recent results identifying the relative positions of Brill--Noether loci, and sketch the Brill--Noether picture of the moduli space of curves. This is based on joint work with Asher Auel, Andrei Bud, Andreas Knutsen, Hannah Larson, and Montserrat Teixidor i Bigas. 

Anda Tenie (Harvard University): Reider-Type Theorems on Normal Surfaces via Bridgeland Stability

Reider’s theorem gives effective criteria for the global generation and very ampleness of adjoint line bundles on smooth surfaces. Reider proves this by applying Bogomolov’s inequality to study the stability of a certain rank-two vector bundle. I will begin by recalling the main ideas of this proof and explaining how Arcara–Bertram reinterpreted it in terms of Bridgeland stability conditions. I will then discuss recent work in which we extend these techniques to obtain Reider-type results on normal projective surfaces, using Langer’s construction of stability conditions in this setting. Our results also hold in positive characteristic and when the dualizing sheaf is not a line bundle. This is based on joint work with Anne Larsen.

Eric Larson (Brown University): Normal bundles of rational curves in Grassmannians

Let $C$ be a general rational curve of degree $d$ in a Grassmannian $G(k, n)$. The natural expectation is that its normal bundle is balanced, i.e., isomorphic to $\bigoplus O(e_i)$ with all $|e_i - e_j| \leq 1$. In this talk, I will describe several counterexamples to this expectation, propose a suitably revised conjecture, and describe recent progress towards this conjecture.