Algebraic Geometry Seminar
Spring 2022 — Tuesdays 3:30  4:30 PM
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Date  Speaker  Title — click for abstract (if available) 
January 25 Virtual @ 11am 
Quentin Posva EPFL 
Gluing theory for slc varieties in positive characteristic
Gluing theory describes the lc pairs that arise as normalisations of slc varieties, and allow to navigate between these two types of objects. In characteristic zero the theory was developed by Kollár, and it has applications to MMP and moduli theory. In this talk, I will review my recent work on gluing theory in positive and mixed characteristics. I will concentrate on surfaces, and if time permits I will sketch the threefold case.

February 1 Virtual 
Sebastian Bozlee Tufts University 
A classification of Gorenstein compactifications of M_{1,n}
The moduli space of pointed algebraic curves M_{g,n} of genus g is of fundamental interest in algebraic geometry. However, it is not compact, so one searches for compactifications of M_{g,n} which are themselves moduli spaces. Each such compactification is a richly interpretable birational model of M_{g,n}.
In this talk we will present a new family of compactifications of M_{1,n} whose points parametrize Gorenstein curves coming from joint work with Adrian Neff and Bob Kuo. These new moduli spaces provably exhaust such Gorenstein modular compactifications of M_{1,n}, the first general classification of modular compactifications since 2012. Time permitting, we will explore the tropical and log geometric reasoning that led to the discovery of these spaces and our classification result.

February 8 Virtual @ 11am 
Samouil Molcho ETH Zurich 
Compactifications of the Universal Jacobian via Tropical Geometry
The Jacobian of the universal curve over the moduli space of stable curves is not proper. The problem of compactifying it has been the subject of long and extensive study. In parallel, an analogous theory has emerged in the realm of tropical geometry, and it has been clear that the two theories are tightly connected. In this talk, I will review the main ideas from the study of compactified Jacobians  the generalized Jacobian and stability conditions on line bundles on families of curves  and some of the tropical analogues. Then, I will discuss precisely how the two theories are connected, and how this connection allows one to translate tropical results into algebraic terms. As a consequence, I will discuss how one can obtain novel compactifications via purely tropical methods.
This is based on joint work with M.Melo, M.Ulirsch, F.Viviani, J. Wise.

February 15 LCB 323 
Andres Fernandez Herrero Cornell University 
Intrinsic construction of moduli spaces via affine Grassmannians
Moduli spaces arise as a geometric way of classifying objects of interest in algebraic geometry. For example, there exists a quasiprojective moduli space that parametrizes stable vector bundles on a smooth projective curve C. In order to further understand the geometry of this space, Mumford constructed a compactification by adding a boundary parametrizing semistable vector bundles. If the smooth curve C is replaced by a higher dimensional projective variety X, then one can compactify the moduli problem by allowing vector bundles to degenerate to an object known as a "torsionfree sheaf". Gieseker and Maruyama constructed moduli spaces of semistable torsionfree sheaves on such a variety X. More generally, Simpson proved the existence of moduli spaces of semistable pure sheaves supported on smaller subvarieties of X. All of these constructions use geometric invariant theory (GIT).
For a projective variety X, the moduli problem of coherent sheaves on X is naturally parametrized by an algebraic stack M, which is a geometric object that naturally encodes the notion of families of sheaves. In this talk I will explain a GITfree construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. Our main technical tools are the theory of Thetastability introduced by HalpernLeistner, and some recent techniques developed by Alper, HalpernLeistner and Heinloth. In order to apply these results, one needs to prove some monotonicity conditions for a polynomial numerical invariant on the stack. We show monotonicity by defining a higher dimensional analogue of the affine grassmannian for pure sheaves. If time allows, I will also explain some applications of these ideas to other moduli problems. This talk is based on joint work with Daniel HalpernLeistner and Trevor Jones. 
February 22 LCB 323 
Giulia Saccà Columbia University 
Moduli spaces on K3 categories are Irreducible Symplectic Varieties
Recent developments by Druel, GrebGuenanciaKebekus, HoringPeternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyperKahler manifolds.
In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold (or of a Gushel Mukai four or six fold) with respect to a generic stability condition are always projective irreducible symplectic varieties. I will rely on the recent work of BayerLahozMacriNeuerPerryStellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

March 1 LCB 323 
Kenneth Ascher UC Irvine 
Kstability and binational models of the moduli space of quartic K3 surfaces
Using Kmoduli we describe various compactifications of the moduli space of quartic K3 surfaces. These compactifications naturally interpolate between the GIT and BailyBorel compactifications, and as a result, we verify LazaO’Grady’s conjecture on the HassettKeelLooijenga program for this moduli space.

March 15 LCB 323 
Jaroslaw Wlodarczyk Purdue 
Functorial resolution by torus actions
We show a simple and fast embedded resolution of varieties and
principalization of ideals in the language of torus actions on ambient smooth
schemes with or without SNC divisors. The canonical functorial resolution of
varieties in characteristic zero is given by, the introduced here, operations
of cobordant blowups with smooth weighted centers. The centers are defined by
the geometric invariant measuring the singularities on smooth schemes with SNC
divisors.
As the result of the procedure we obtain a smooth variety with a torus action
and the exceptional divisor having simple normal crossings. Moreover, its
geometric quotient is birational to the resolved variety, has abelian quotient
singularities, and can be desingularized directly by combinatorial methods. The
paper is based upon the ideas of the joint work with Abramovich and Temkin and
a similar result by McQuillan on resolution in characteristic zero via
stacktheoretic weighted blowups.
As an application of the method we show the resolution of a certain class of
isolated singularities in positive and mixed characteristic.

March 22 Virtual 
Isabel Vogt Brown University 
Normal bundles of canonical curves
The extrinsic geometry of the canonical model of a nonhyperelliptic curve captures many aspects of the intrinsic geometry of the curve. In this talk I will discuss joint work with Izzet Coskun and Eric Larson in which we show that the normal bundle of a general canonical curve of genus at least 7 is always semistable. This makes substantial progress towards a conjecture of AproduFarkasOrtega, and answers it completely in a third of all cases.

March 29 LCB 323 
Fanjun Meng Northwestern University 
Kodaira dimension of fibrations over abelian varieties
The Kodaira dimension of smooth projective varieties is an important birational invariant. In this talk, we will discuss some conjectures on the behavior of Kodaira dimension proposed by Popa. We prove an additivity result for the log Kodaira dimension of algebraic fiber spaces over abelian varieties, a superadditivity result for algebraic fiber spaces over varieties of maximal Albanese dimension, as well as a subadditivity result for log pairs over abelian varieties. This is joint work with Mihnea Popa.

April 5 LCB 323 
Tommaso De Fernex University of Utah 
Jet fibers of quasifinite morphisms
Any morphism of schemes over a field induces a morphism at the level of arc spaces. We will discuss some finiteness results about the fibers of the latter, and present various applications to arc spaces and beyond.

April 12 Virtual 
 
Rescheduled
Rescheduled

April 19 Virtual 
Joe Waldron Michigan State University 
Purely inseparable Galois theory
Given a field \(K\) of characteristic \(p\), a classical result of Jacobson provides a Galois correspondence between finite purely inseparable subfields of exponent one (those with \(K^p\subset L\subset K\)), and subrestricted Lie algebras of \(\mathrm{Der}(K)\). I will discuss joint work with Lukas Brantner in which we extend this Galois correspondence to subfields of arbitrary exponent using methods from derived algebraic geometry.

April 26 Virtual 
Hunter Dinkins UNC Chapel Hill 
Exotic Quantum Difference Equations and Integral Solutions
From their beginning, Nakajima quiver varieties have bridged the gap between algebraic geometry and representation theory. Maulik and Okounkov pioneered new such directions in 2012, as they constructed a Hopf algebra acting in the cohomology of quiver varieties. One of their main results is the identification of the quantum differential equation governing the enumerative geometry of quiver varieties with operators in this Hopf algebra. A similar identification was obtained later by Okounkov and Smirnov for the Ktheoretic difference equations. Their results suggest a natural generalization of the difference equations, which will be the focus of this talk. I will define these new difference equations and demonstrate how their solutions can be related back to enumerative geometry. In the case that the quiver variety has finitely many torus fixed points, these results provide integral solutions for these difference equations and are related to the algebraic Bethe Ansatz. I will give examples of all results for the Hilbert scheme of points in the plane.

May 3 Virtual 
David Favero University of Alberta 
Homotopy Path Algebras
I will consider a class of algebras, “Homotopy Path Algebras”, naturally appearing in many contexts; e.g. algebraic topology, sheaf theory, and toric geometry (as full strong exceptional collections of line bundles). I will develop the general theory of such algebras and explain the connection to homological mirror symmetry, expanding on the ideas of BondalRyan and FanLuiTruemannZaslow. This is based on joint work with Jesse Huang.

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