August 23: Izzet Coskun (University of Illinois Chicago)
Location: Zoom (Special Time: 4:00-5:00PM EDT)
Title: The stability of normal bundles of curves
Abstract: Given a smooth projective curve X, there are several naturally defined vector bundles on X such as the normal bundle of X or the restriction of the tangent bundle of projective space to X. Similarly given a branched covering of curves f: Y → X, there are several natural vector bundles such as the Tschirnhausen bundle on X associated with the covering. In this talk, I will discuss the stability of normal bundles of Brill-Noether curves and Tschirnhausen bundles. The talk is based on joint work with Eric Larson, Isabel Vogt and Geoffrey Smith.
August 30: Junyan Zhao (University of Illinois Chicago)
Location: UNIV 301
Title: Hassett-Keel program and K-stability
Abstract: The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six.
Pre-talk: 2:30 pm at Math Building 731, about geometry of del Pezzo surfaces and moduli spaces of DM-stable curves.
September 6 and September 13: Jarosław Włodarczyk (Purdue University)
Location: UNIV 301
Title: Functorial resolution by torus actions
Abstract: We present a straightforward and efficient method for embedded resolution of varieties and the principalization of ideals, expressed in the language of torus actions on smooth ambient varieties with simple normal crossings (SNC) divisors.
The canonical functorial resolution of varieties in characteristic zero is obtained by the operations of cobordant blow-ups with smooth weighted centers. These centers are determined by a geometric invariant that measures the singularities on smooth schemes with SNC divisors.
As a 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.
We also extend the construction of cobordant blow-ups to more general class of the transformations using Cox rings associated with proper birational morphisms.
As an application of the method, we show the resolution of certain classes of singularities in positive and mixed characteristic.
The talk is based upon my two recent papers "Resolution by torus actions" and "Cox rings of morphisms and resolution of singularities" and the ideas of the joint project with Abramovich and Temkin and a similar result by McQuillan on resolution in characteristic zero via stack-theoretic weighted blow-ups.
This is a two part talk. Part I is on September 6 and Part II is on September 13.
September 20: Emelie Arvidsson (University of Utah)
Location: Zoom (Special Time: 12:30-1:30PM EDT)
Title: Properties of log canonical singularities in positive characteristic
Abstract: We will investigate if some well known properties of log canonical singularities over the complex numbers still hold true over perfect fields of positive characteristic and over excellent rings with perfect residue fields. We will discuss both pathological behavior in characteristic p as well as some positive results for threefolds. We will see that the pathological behavior of these singularities in positive characteristic is closely linked to the failure of Kodaira-type vanishing theorems in positive characteristic. Additionally, we will explore how these questions are related to the moduli theory of varieties of general type.
This is based on joint work with F. Bernasconi and Zs. Patakfalvi, as well as joint work with Q. Posva.
September 27: Renjie Lyu (Morningside Center of Mathematics)
Location: Zoom (Special Time: 9:00-10:00AM EDT)
Title: Degeneration of Hodge structures and cubic hypersurfaces
Abstract: The degeneration of Hodge structures is related to how a smooth projective variety degenerate. And it provides a Hodge-theoretic perspective to compactify moduli spaces. In this talk, I will focus on a particular degeneration of cubic hypersurfaces and study the associated limiting mixed Hodge structure. It generalizes some results in Radu Laza’s and Brendan Hassett’s works on cubic fourfolds. This is a joint work with Zhiwei Zheng.
Pre-talk: (8:20-8:50AM on Zoom): Examples of (Mixed) Hodge structures on algebraic varieties.
Pre-talk Abstract: On the cohomology group of a compact Kähler manifold, a Hodge structure can be attached. It is an algebraic structure which refines the cohomology group. In this pre-talk, I will introduce the notion of Hodge structures through simple examples like curves, surfaces, etc. I will also talk about how the mixed Hodge structures naturally arise in geometric ways.
October 4: Takumi Murayama (Purdue University)
Location: UNIV 301
Title: The relative minimal model program for excellent schemes, algebraic spaces, and analytic spaces
Abstract: In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications.
In this talk, I will discuss joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces that includes formal schemes, rigid analytic spaces, Berkovich spaces, and adic spaces, all in equal characteristic zero. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤3 and ≤2, respectively. Our results for formal schemes, Berkovich spaces, and adic spaces are completely new.
Pre-Talk (2:30PM in MATH 731): Which complex manifolds can be parametrized?
Abstract: Let X be a closed complex hypersurface in CP^n. If X is defined by a linear equation, then X is isomorphic to CP^{n-1}, and can therefore be parametrized on an open dense subset by C^n. If X is defined by a quadratic equation, then X can be parametrized in a similar fashion via stereographic projection. A fundamental question in birational geometry (a subfield of algebraic geometry) is when closed complex submanifolds of CP^n can be parametrized in this fashion. In this talk, we will introduce this question and related questions arising from the classification of algebraic varieties "up to birational equivalence." The approaches to study these questions require an interesting blend of algebra with techniques from complex analysis, topology, and arithmetic.
October 11: No Talk
October 18: Tatsuki Yamaguchi (University of Tokyo)
Location: UNIV 301
Title: Pure subrings of singularities of dense F-pure type
Abstract: Let X, Y be affine varieties over C. If a morphism f:Y→X is pure and Y has good singularities, then so does X, e.g., if Y is rational, log-terminal, Du Bois. Zhuang conjectured that this type of theorem also holds for log canonical singularities. I will discuss this problem for singularities of dense F-pure type, expected to be equivalent to log canonical singularities. In this talk, I will use ultraproducts to define a variant of F-pure singularities in equal characteristic zero.
Pre-Talk (2:30PM in MATH 731): F-pure singularities
Abstract: In algebraic geometry of positive characteristic, some classes of singularities are defined by using Frobenius morphisms. They are closely related to singularities in birational geometry in equal characteristic zero. I will talk about F-pure singularities and reduction modulo p.
October 25: Joe Foster (University of Massachusetts Amherst)
Location: UNIV 301
Title: The Lefschetz standard conjectures for IHSMs of generalized Kummer type
Abstract: For a smooth complex projective variety X, the Lefschetz standard conjectures of Grothendieck predict the existence of certain algebraic cycles on the self-product X\times X. The conjectures have many important implications, including those for the theory of motives, the Hodge conjecture, Hodge homotheties, and the period-index conjecture. Contingent on dimension, I will describe the partial proof of the Lefschetz standard conjectures for irreducible holomorphic symplectic manifolds (IHSMs) of generalized Kummer deformation type. Our strategy relies on Markman's construction of universal families of marked IHSMs, Verbitsky's theory of hyperholomorphic sheaves, and Magni's construction of derived equivalences between certain generalized Kummer varieties.
Pre-Talk (2:30PM in MATH 731): The geometry of hyperkähler varieties.
Abstract: An irreducible holomorphic symplectic manifold, or hyperkähler manifold, is a simply connected compact kähler manifold that carries a unique symplectic 2-form. In this pretalk I will describe the construction of the known deformation types of these objects, as well as their unique geometry.
November 1: Abhishek Oswal (Michgan State University)
Location: UNIV 301 (joint with Model theory seminar)
Title: p-adic Borel hyperbolicity of Shimura varieties of abelian type
Abstract: Let S be a Shimura variety such that every connected component of the space of complex points of S arises as the quotient of a Hermitian symmetric domain by a torsion-free arithmetic group. In the 1970s, Borel proved that any holomorphic map from a complex algebraic variety V into such a Shimura variety S is algebraic. In this talk, I'll discuss joint work with Anand Patel, Ananth Shankar, and Xinwen Zhu on a p-adic version of this result.
November 8: Peter McDonald (University of Utah)
Location: UNIV 301
Title: Multiplier ideals and klt singularities via (derived) splittings
Abstract: Thanks to the Direct Summand Theorem, splinter conditions have emerged as a way of studying singularities in commutative algebra and algebraic geometry. In characteristic zero, work of Kovács (2000) and Bhatt (2012) characterizes rational singularities as derived splinters. In this talk, I will present an analogous characterization of klt singularities by imposing additional conditions on the derived splinter property. This follows from a new characterization of the multiplier ideal, an object that measures the severity of the singularities of a variety, viewing it as a sum of trace ideals. This perspective also gives rise analogous description of the test ideal in characteristic as a corollary to a result of Epstein-Schwede (2014).
November 15: Sujatha Ramdorai (University of British Columbia)
Location: UNIV 301 (Note: The speaker will also give a Colloquium talk on Nov. 14)
Title: Modules over Iwasawa algebras and Krull dimension
Abstract: Let $p$ be an odd prime and let $$\mathbb{Z}_p$$ the ring of $p$-adic integers. The power series rings $$\Lambda_d := \mathbb{Z}_p[[T_1,T_2,\cdots, T_d]]$$ arise naturally in the study of arithmetic of elliptic curves over number fields. In particular, certain finitely generated modules over such rings play an important role in arithmetic and there are conjectures on the Krull dimensions of these modules. In this talk, we shall explain the interactions between Arithmetic and Commutative Algebras in this context.
November 29: Donu Arapura (Purdue University)
Location: UNIV 301
Title: Differential forms on singular varieties
Abstract: Since this is “in house”, rather than give a polished talk, I thought I'd talk about something that I don’t understand very well. Given a singular complex variety X, there are a couple of reasonable substitutes for the sheaf of p-forms. One is the direct image of the p-forms from a resolution of singularities, another is a more sophisticated variant due to Du Bois. There is a map from one to the other. I conjecture that X has rational singularities if and only it is normal and these maps are surjective for all p. I know how to prove one direction when X is also projective. To go further, I would need the existence of a partial resolution of singularities is a suitable sense. I have another conjecture about the what happens to these for GIT quotients.
Pre-Talk (2:30PM in MATH 731): Kähler differentials on algebraic varieties.
Abstract: I want to explain how one introduces differential forms into algebraic geometry and commutative algebra. I will mostly focus on examples which illustrates both the good and the bad properties of the construction.
December 6: Lisa Marquand (Courant Institute at New York University)
Location: UNIV 301
Title: The defect of a cubic threefold
Abstract: The defect of a cubic threefold with isolated singularities is a measure of the failure of Poincare duality, and also of the failure to be Q-factorial. From the work of Cheltsov, a cubic threefold with only nodal singularities is Q factorial if and only if there are at most 5 nodes. We investigate the defect of cubic threefolds with worse singularities, and provide a geometric method to compute this global invariant. One can then compute the Mixed Hodge structure on the middle cohomology of the cubic fourfold, in terms of the defect and local invariants of the singularity types. We then relate the defect to geometric properties of the cubic threefold, showing it is positive if and only if the cubic contains a plane or a rational normal cubic scroll. This is joint work with Sasha Viktorova.
Pre Talk (2:30PM in MATH 731): Classical geometry of cubic hypersurfaces
Abstract: In this talk, we will discuss some geometric constructions that we can do from cubic hypersurfaces. In particular, we will exhibit various structures that (possibly singular!) cubic threefolds and fourfolds exhibit by projection from linear subspaces.