Algebraic Geometry Seminar
Fall 2026
Regular Day: Tuesday
Regular Time: 3:30PM - 4:30PM
Regular Location: LCB 215
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August 25 |
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September 1 |
Yuchen Liu
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The notion of special valuations, i.e. finitely generated valuations inducing klt degenerations of Fano
varieties, plays an important role in the recent study of K-stability and moduli of Fano varieties. Given an affine
log Calabi-Yau variety $U$, we show that its special skeleton, namely the space of special valuations in the dual
complex of $U$, is independent of the choice of its Fano compactifications. Consequently, the special skeleton of
$U$ is invariant under the action of $\text{Aut}(U)$. As an example, when $U$ is the complement of a Markov cubic surface
in the affine three-space, we show that the action of three Vieta involutions on the special skeleton exhibits a
surprising connection with dynamics on the hyperbolic plane.
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September 15 |
Sridhar Venkatesh
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For a complex algebraic variety X embedded inside a smooth variety Y, the local cohomology sheaves of X in Y carry additional structure of a (mixed) Hodge module. In the hypersurface and the
local complete intersection (lci) case, this has been widely leveraged to prove various results about higher Du Bois and higher rational singularities, among other things. We investigate these local
cohomology sheaves when X is a toric variety (which is typically non-lci) and prove various results about them. A few applications include showing that the local cohomological dimension of a toric
variety is NOT a combinatorial invariant, and some new results about the singular cohomology of toric varieties. This is based on joint work with Hyunsuk Kim.
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September 22 |
Jonathon Fleck
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The Hilbert schemes classifying the closed subschemes of projective
space with a fixed Hilbert polynomial are normally thought of as static
objects via Grothendieck's construction. Moreover, for projective spaces of
dimensions three or more they have been shown to satisfy a Murphy's Law of
arbitrarily bad behavior. The Hilbert schemes are known, however, to be
linearly connected via Borel-fixed ideals by a Theorem of Hartshorne. In
this talk, we use Borel-fixed ideals to realize the Hilbert schemes as
moduli spaces of Bridgeland-stable objects and use that to study variational
properties.
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September 29 |
Fabio Bernasconi
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A folklore question to arithmetic geometry (attributed to Shafarevich and Grothendieck) asks what are the
smooth projective varieties of dimension (d) over (\operatorname{Spec}(\mathbb Z)). In relative dimensions
(0) and (1), a complete answer is known, thanks to the work of Minkowski, Gauss, and Abrashkin–Fontaine,
which I will briefly recall. Starting in relative dimension (2), the problem remains open. I will discuss
some of the known restrictions and present a complete classification in the case of surfaces of Kodaira
dimension at most 0, obtained in joint work with G. Martin and Zs. Patakfalvi. If time permits, I will
also explain some examples and phenomena in higher dimensions.
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October 6 |
Daniel Apsley
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Weighted K-stability is a generalization of K-stability that incorporates an algebraic torus action and is
closely related to the existence of Kähler–Ricci soliton metrics on complex Fano manifolds. Recent work has
established a proper moduli theory of weighted K-polystable Fano varieties. In contrast with ordinary
K-stability, every Fano variety admits a degeneration to a weighted K-polystable variety, allowing these
moduli spaces to capture a substantially broader class of Fano varieties. I will conclude by discussing the
construction of ample \mathbb{R}-line bundles on weighted K-moduli spaces.
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October 20 |
Ying Wang
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Coming soon
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October 20 special time 4:30pm LCB 215 |
Jefferson Baudin
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Informally, a variety is "irregular" if it is closely related to an abelian variety
(that is, a smooth projective variety which also admits the structure of a group).
This is for example the case of non-rational curves, which embed in their Jacobian.
Over the complex numbers, several methods gave rise to remarkable results in the
understanding of these varieties: characterization of abelian varieties by only
fixing a few invariants, deeper understanding of the Euler characteristic of
irregular varieties, study of their pluricanonical systems, and so on (in any
dimension!). These theorems rely on analytic techniques, making this whole topic
harder to reach in positive characteristic. Our goal in this talk will be to explain
purely positive characteristic methods that allow us to "approximate well enough the
complex theory", in order to deduce geometric consequences - a program initiated by
Hacon and Patakfalvi. We will achieve this through presenting the following theorem:
if X is a smooth projective ordinary variety of maximal Albanese dimension (i.e.
dim(alb(X)) = dim(X)), then the Euler characteristic of the sheaf of top forms is
non-negative. If in addition this quantity is zero, then the Albanese image of X is
fibered by abelian varieties. The proof uses varied tools, such as a Witt vector
version of Grauert-Riemenschneider vanishing.
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October 27 |
Dano Kim
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Coming soon
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November 3 |
Roi Docampo
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Coming soon.
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November 10 |
Franco Rota
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Coming soon
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December 1 |
Anh Duc Vo
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Coming soon
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December 8 |
Wanchun Shen
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Coming soon
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