Algebraic Geometry Seminar

Spring 2024 — Tuesdays 3:30 - 4:30 PM

LCB 222

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Date Speaker Title — click for abstract (if available)
January 16
At 3 PM
Junyan Zhao
University of Illinois Chicago
Moduli of curves and K-stability
In this talk, I will explore various compactifications of moduli spaces of genus six curves, delving into their intricate interactions. A key ingredient is the K-moduli spaces of log Fano pairs. The intricate wall crossing structure not only yields novel compactifications but also plays a pivotal role in understanding the Hassett-Keel program of moduli spaces of curves.
January 23
January 30
February 6 Brad Dirks
Stony Brook
The Minimal Exponent of LCI Subvarieties
Classification of singularities is an interesting problem in many areas of algebraic geometry, like the minimal model program. One classical approach is to assign to a singular subvariety a rational number, its log canonical threshold. For complex hypersurface singularities, this invariant has been refined by M. Saito to the minimal exponent. This invariant is related to Bernstein-Sato polynomials, Hodge ideals and higher du Bois and higher rational singularities. In joint work with Qianyu Chen, Mircea Mustață and Sebastián Olano, we defined the minimal exponent for LCI subvarieties of smooth complex varieties. We related it to local cohomology, higher du Bois and higher rational singularities. I will describe what was done in the hypersurface case, give our definition in the LCI case and explain the relation to local cohomology modules and the classification of singularities.
February 13
February 20 Gabriel Dorfsman-Hopkins
St. Lawrence University
A Condensed Approach to Continuous Group Cohomology
A common issue one can run into is that topological algebra doesn't play well with homological algebra: for example, the category of topological abelian groups is not an abelian category. This introduces many pathologies, for example: continuous group cohomology does not give rise to long exact sequences. Clausen and Scholze (resp, Barwick and Haine) suggest a solution to this kind of problem by extending the category of topological abelian groups to the abelian category of condensed abelian groups (resp. pyknotic abelian groups), which are sheaves of abelian groups on the pro-etale site of a geometric point. We will explain a construction of continuous group cohomology in this condensed setting, give comparisons to classical continuous group cohomology, and explain some consequences and future directions that fall out of this formalism (including, if time allows, to p-adic and perfectoid geometry).
February 27
Iacopo Brivio
Harvard
Anti-Iitaka conjecture in positive characteristic
A famous conjecture by Iitaka predicts that, if \(f\colon X\to Y\) is a fibration of smooth projective varieties over \(\mathbb{C}\) with general fiber \(F\), then \(\kappa(K_X)\geq \kappa(K_F)+\kappa(K_Y)\). It was recently shown by Chang that a similar inequality holds for the anticanonical divisors: when the stable base locus \(\mathbf{B}(-K_X)\) is vertical, then \(\kappa(-K_X)\leq \kappa(-K_F)+\kappa(-K_Y)\). Both the Iitaka conjecture and Chang's theorem are known to be false over fields of positive characteristic. However the expectation is that these inequalities should still hold whenever \(F\) is sufficiently well-behaved with respect to the Frobenius morphism. In this talk I will explain how to recover Chang's theorem for this kind of tame fibrations in characteristic \(p>0\) and discuss some related questions. This is based on joint work with M. Benozzo and C.K. Chang.
March 5 No talk (spring break)
March 12 Bogdan Zavyalov
IAS
March 19
March 26 Joaquín Moraga
UCLA
April 2
April 9
Fred Nelson The Group of Rational Points on the Holm Curve is Torsion-free
April 16

Archive of previous seminars.


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