Algebraic Geometry Seminar

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.

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.

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.

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 "torsion-free sheaf". Gieseker and Maruyama constructed moduli spaces of semistable torsion-free 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 GIT-free 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 Theta-stability introduced by Halpern-Leistner, and some recent techniques developed by Alper, Halpern-Leistner 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 Halpern-Leistner and Trevor Jones.

Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell 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 hyper-Kahler 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 Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

Using K-moduli we describe various compactifications of the moduli space of quartic K3 surfaces. These compactifications naturally interpolate between the GIT and Baily-Borel compactifications, and as a result, we verify Laza-O’Grady’s conjecture on the Hassett-Keel-Looijenga program for this moduli space.

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 blow-ups 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 stack-theoretic weighted blow-ups. As an application of the method we show the resolution of a certain class of isolated singularities in positive and mixed characteristic.

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 Aprodu--Farkas--Ortega, and answers it completely in a third of all cases.

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.

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.

Rescheduled

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 sub-restricted 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.

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 K-theoretic 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.

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 Bondal-Ryan and Fan-Lui-Truemann-Zaslow. This is based on joint work with Jesse Huang.