Schedule of Talks: Autumn 2021

Time/Location : Seminar will be on Tuesdays 3:00-4:00pm via Zoom and occasionaly in person in Mw 154


 

TIME SPEAKER TITLE HOST
September 7 
T  
 
Michel Brion, CNRS and Institut Fourier Homomorphisms of algebraic groups: representability and rigidity Joshua
October 19 
T  
 
Padma Srinivasan (University of Georgia) Some Galois cohomology classes arising from the fundamental group of a curve Joshua
November 2 

 
Open
November 16 

 
Roberto Pirisi (University of Rome, Italy) Brauer groups of moduli of hyperelliptic curves and their compactifications, via cohomological invariants Joshua
November 30 
T  
 
Open


Schedule of Talks: Spring

Time/Location : Seminar will be on Tuesdays 3:00-4:00pm via Zoom and occasionaly in person in Mw 154


 

TIME  SPEAKER TITLE HOST
January 18 
T  
 
Open
February 1 
T  
 
Open
February 15 
T  
 
Open
February 22 
T  
 
Agneet Dhillon (University of Western Ontario)
March 8 
T  
 
Open
March 29 
T  
 
Open
April 12 
T  
 
Open
April 26 
T  
 
Open

Abstracts

(Brion Talk) The topic of the talk is the functor of homomorphisms from G to H, where G and H are algebraic groups over a field k. When G is linearly reductive and H is smooth, we show that this functor is represented by a smooth scheme M; moreover, every orbit of H acting by conjugation on M is open. (The notion of linear reductivity will be discussed in detail during the talk). This result extends a theorem of Grothendieck in which G is of multiplicative type, and has applications to algebraic group actions on projective varieties.

Zoom details for this seminar: https://osu.zoom.us/j/91057354055?pwd=RlFTTCt5SFlDS3VKV01LWnJodlkwQT09 Meeting id: 910 5735 4055, Passcode: 033969

(Srinivasan talk) We will talk about a few Galois cohomology classes naturally arising from the fundamental group of a curve.

We will first talk about the Ceresa class, which is the image under a cycle class map of a canonical algebraic cycle associated to a curve in its Jacobian. This class vanishes for all hyperelliptic curves and was expected to be nonvanishing for non-hyperelliptic curves. In joint work with Dean Bisogno, Wanlin Li and Daniel Litt, we construct a non-hyperelliptic genus 3 quotient of the Fricke-Macbeath curve with vanishing Ceresa class, using the character theory of the automorphism group of the curve, namely, PSL_2(F_8).

Time permitting, we will also talk about some Galois cohomology classes that obstruct the existence of rational points on curves, by obstructing splittings to natural exact sequences coming from the fundamental group of a curve. In joint work with Wanlin Li, Daniel Litt and Nick Salter, we use these obstruction classes to give a new proof of Grothendieck’s section conjecture for the generic curve of genus g > 2. An analysis of the degeneration of these classes at the boundary of the moduli space of curves, combined with a specialization argument lets us produce infinitely many curves of each genus over p-adic fields and number fields that satisfy the section conjecture.

(Pirisi talk) Given an algebraic variety X, the Brauer group of X is the group of Azumaya algebras over X, or equivalently the group of Severi-Brauer varieties over X, i.e. fibrations over X which are étale locally isomorphic to a projective space. It was first studied in the case where X is the spectrum of a field by Noether and Brauer, and has since became a central object in algebraic and arithmetic geometry, being for example one of the first obstructions to rationality used to produce counterexamples to Noether's problem of whether given a representation V of a finite group G the quotient V/G is rational. While the Brauer group has been widely studied for schemes, computations at the level of moduli stacks are relatively recent, the most prominent of them being the computations by Antieau and Meier of the Brauer group of the moduli stack of elliptic curves over a variety of bases, including Z, Q, and finite fields.

In a recent series of joint works with A. Di Lorenzo, we use the theory of cohomological invariants, and its extension to algebraic stacks, to completely describe the Brauer group of the moduli stacks of hyperelliptic curves, and their compactifications, over fields of characteristic zero, and the prime-to-char(k) part in positive characteristic. It turns out that the Brauer group of the non-compact stack is generated by elements coming from the base field, cyclic algebras, an element coming from a map to the classifying stack of étale algebras of degree 2g+2, and when g is odd by the Brauer-Severi fibration induced by taking the quotient of the universal curve by the hyperelliptic involution. This paints a richer picture than in the case of elliptic curves, where all non-trivial elements come from cyclic algebras. Regarding the compactifications, there are two natural ones, the first obtained by taking stable hyperelliptic curves and the second by taking admissible covers. It turns out that the Brauer group of the former is trivial, while for the latter it is almost as large as in the non-compact case, a somewhat surprising difference as the two stacks are projective, smooth and birational, which would force their Brauer groups to be equal if they were schemes.

  • Arithmetic Geometry Seminar calendar from 2020-2021
  • Arithmetic Geometry Seminar calendar from 2019-2020


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