Algebraic Geometry Seminar
Fall 2022 — Tuesdays 3:30  4:30 PM
LCB 323
Join the Algebraic Geometry mailing list for updates + announcements.Date  Speaker  Title — click for abstract (if available) 
August 31st @ 2pm 
Matt Baker Georgia Tech 
Nonarchimedean and tropical geometry, algebraic groups, moduli spaces of matroids, and the field with one element
I will give an introduction to Oliver Lorscheid’s theory of ordered blueprints – one of the more successful approaches to “the field of one element” – and sketch its relationship to Berkovich spaces, tropical geometry, Tits models for algebraic groups, and moduli spaces of matroids. The basic idea for the latter two applications is quite simple: given a scheme over Z defined by equations with coefficients in {0,1,1}, there is a corresponding “blue model” whose Kpoints (where K is the Krasner hyperfield) sometimes correspond to interesting combinatorial structures. For example, taking Kpoints of a suitable blue model for a split reductive group scheme G over Z gives the Weyl group of G, and taking Kpoints of a suitable blue model for the Grassmannian G(r,n) gives the set of matroids of rank r on {1,...,n}. Similarly, the Berkovich analytification of a scheme X over a valued field k coincides, as a topological space, with the set of Tpoints of X, considered as an ordered blue scheme over k. Here T is the tropical hyperfield, and Tpoints are defined using the observation that a (height 1) valuation on k is nothing other than a homomorphism to T.

September 6th 
Joaquín Moraga UCLA 
Complexity and coregularity of Fano varieties
In this talk, we discuss Fano varieties, one of the three building blocks of algebraic varieties.
Among them, the most wellknown class are Fano toric varieties that can be described combinatorially.
In this talk, we will introduce two invariants for Fano varieties: the complexity and coregularity.
Both invariants take value 0 when the Fano variety is toric.
We will explain how these two invariants allow us to detect toric varieties among Fano varieties,
and what is expected for Fano varieties when these invariants are near 0.

September 13th 
Raymond Cheng Columbia University 
qbic Hypersurfaces
Let’s count: 1, 2, q+1. The eponymous objects are special projective hypersurfaces of degree q+1, where q is a power of the positive ground field characteristic. This talk will sketch an analogy between the geometry of qbic hypersurfaces and that of quadric and cubic hypersurfaces. For instance, the moduli spaces of linear spaces in qbics are smooth and themselves have rich geometry. In the case of qbic threefolds, I will describe an analogue of result of Clemens and Griffiths, which relates the intermediate Jacobian of the qbic with the Albanese of its surface of lines.

September 20th 
Lei Wu KU Leuven 
Logarithmic cotangent bundles, Chern classes, and applications
Using MacPherson's Euler obstruction function, one can identify the abelian group of constructible functions with the group of algebraic cycles on a smooth complex algebraic variety. Kashiwara's local index formula gives an alternative approach to this identification by using characteristic cycles for holonomic Dmodules (they are Lagrangian cycles in the cotangent bundle). This identification then enables us to define Chern classes of algebraic cycles by using characteristic cycles. In this talk, I will first explain how to obtain Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement by using logarithmic cotangent bundles motivated by Dmodule theory. Then I will discuss applylications of such Chern classes in understanding ChernMather classes of very affine varieties and in proving the Involution Conjecture of Huh and Sturmfels in likelihood geometry. This work is joint with Maxim, Rodriguez, and Wang.

September 27th 
Pat Lank University of South Carolina 
Generation of derived categories in prime characteristic
Within this talk, which is joint work with Matthew Ballard, we will study the bounded derived category of coherent sheaves $D^b (\operatorname{coh}X)$ on an $F$finite Noetherian scheme of prime characteristic $p$. It will be shown that $F_\ast^e (\operatorname{perf}X)$ strongly generates $D^b (\operatorname{coh}X)$ for $e\gg 0$ where $F_\ast^e$ denotes the $e$th iterate of the Frobenius pushforward functor on $X$. Immediate applications to this is a new proof of Kunz's theorem, introduces new invariants for singularity types in prime characteristic coming from derived categories, categorical resolution of singularities, and produces interesting structural insight into this category. This motivates a notion of $F$thickness; namely, when $F_\ast^e \mathcal{O}_X$ strongly generates $D^b (\operatorname{coh}X)$. Lastly, in the case of smooth Fano projective complete intersections $X$, it will be shown that $F_\ast^e \mathcal{O}_X$ decomposes into direct summands indexed by components in Kuznetsov/Orlov style semiorthogonal decompositions. Particularly, when $p$ is sufficiently large, $X$ is globally $F$split.

October 4th 
Rachel Webb UC Berkeley 
Hasset moduli stacks of twisted curves
A stable nmarked curve is a nodal curve with n distinct marked points and finitely many automorphisms. If we choose rational numbers a_1, . . ., a_n in the interval (0, 1], then a weighted stable nmarked curve is a generalization where the marks are allowed to coincide as long as the total weight at any point is at most one. Moduli of weighted stable curves were first constructed by Hassett. On the other hand, a twisted stable nmarked curve is a tame stack whose coarse moduli space is a stable nmarked curve, such that stacky structure is concentrated at nodes and markings and has a specific local description. I will discuss a modification (using log geometry) of the moduli of twisted stable curves where the markings are allowed to coincide, analogous to Hassett's construction for representable curves. This is a joint project with Martin Olsson.

October 18th 
Yueqiao Wu University of Michigan 
A nonArchimedean characterization of local Kstability
Log Fano cones are generalizations of cones over log Fano pairs, and have a local Kstability theory extending the one for log Fano pairs. In this talk, we aim to give a characterization for local Kstability from a nonArchimedean point of view, which will allow us to study general test configurations without putting strong singularity assumptions on them. As a consequence of this characterization, we will see that one direction of the YTD conjecture for log Fano cones, as shown earlier by CollinsSzékelyhidi and Li, can be strengthened.

October 19th 
Lena Ji University of Michigan 
Fano hypersurfaces with no finite order birational automorphisms
We show that a very general ndimensional complex hypersurface X of degree ≥ 5⌈(n+3)/6⌉ has no finite order birational automorphisms. In particular, this applies to higher index Fano hypersurfaces (of sufficiently high dimension). We prove this result by studying the specialization homomorphism for the birational automorphism group. This work is joint with Nathan Chen and David Stapleton.

October 25th 
Mahrud Sayrafi University of Minnesota 
Short resolutions of the diagonal and a Horrockstype splitting criterion in Picard rank 2
In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Motivated by the study of indecomposable vector bundles, in 1978 Beilinson constructed a resolution of the diagonal on P^n which has been used to great effect in algebraic geometry. We obtain a Horrockstype splitting criterion (under an additional hypothesis) for vector bundles over a smooth projective toric variety X of Picard rank 2 using a linear resolution of the diagonal consisting of finite direct sums of line bundles. Since this resolution has length dim(X), we also prove a new case of a conjecture of BerkeschErmanSmith that predicts a version of Hilbert's Syzygy Theorem for virtual resolutions. This is joint work with Michael Brown.

November 1st 
Daniel Bragg University of Utah 
A Stacky Murphy’s Law for the Stack of Curves
We show that every DeligneMumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich.

November 8th 
Justin Lacini University of Kansas 
Syzygies of adjoint linear series on projective varieties
Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In joint work with Purnaprajna Bangere we give a positive answer to this question.

November 18th 
Philip Engel University of Georgia 
Elliptic surfaces and a Torelli theorem
Elliptic surfaces fibering over an elliptic curve with 12 singular fibers form a tendimensional moduli space. Their middle cohomology has a K3 type Hodge structure endowing the moduli space with a period map into a tendimensional arithmetic quotient. I will discuss a proof that this period map is dominant (like moduli of K3 surfaces) but that the degree of the period map exceeds one (unlike moduli of K3 surfaces). This is joint work with F. Greer and A. Ward.

November 29th 
Emelie Arvidsson University of Utah 
Properties of logcanonical threefold singularities in positive characteristics
In this talk I will study if some well—known properties of log canonical threefold singularities over the complex numbers hold true over perfect fields of positive characteristics? For surfaces of Fano type the Kawamata–Viehweg vanishing theorem, which in general famously fails in positive characteristic p, hold true if p>5. This has the consequence that threefold klt singularities are Cohen–Macaulay in this characteristic.
With this positive result at hand it is natural to ask if Kollárs theorem, asserting that the depth of a log canonical threedimensional singularity is three at any closed point which itself is not a log canonical center, also can be extended to positive (possibly large enough) characteristics? Another natural and related question is if minimal log canonical centers are normal in (possibly large enough) positive characteristics?
In this talk we will answer these two questions. This includes joint work with Bernasconi–Patakfalvi and Posva.

December 6th 
Kimoi Kemboi Cornell University 
Full strong exceptional collections on ranktwo linear GIT quotients
A full strong exceptional collection is an important structure to have on a derived category with many useful implications. For instance, such a collection produces a basis for the Grothendieck group and a tilting object. In this talk, we will discuss the existence of full strong exceptional collections consisting of vector bundles on certain linear GIT quotients by a split reductive group G of rank two. These vector bundles will come from irreducible Grepresentations whose weights lie in a particular “window” in the weight space of G. This is based on joint work with Daniel HalpernLeistner.

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