Graduate Algebraic Geometry Seminar

Multiplicities of Jumping Numbers

Apr 27, 2022, SEO 636

Speaker:Swaraj Pande, University of Michigan

Abstract: Multiplier ideals are refined invariants of singularities of algebraic varieties. They give rise to other numerical invariants, for example, the log canonical threshold and more generally, jumping numbers. This talk is about another related invariant, namely multiplicities of jumping numbers. For an m-primary ideal I in the local ring of a smooth complex variety, multiplicities of jumping numbers measure the difference between successive multiplier ideals of I. The main result is that these multiplicities naturally fit into a quasi-polynomial. We will also discuss when the various components of this quasi-polynomial have the highest possible degree, relating it to the Rees valuations of I. As a consequence, we derive some formulas for a subset of jumping numbers of m-primary ideals. Time permitting, we will consider the special case of monomial ideals where these invariants have a combinatorial description in terms of the Newton polyhedron.

Reference: Multiplicities of Jumping Numbers, arxiv.

Formally Unramified Algebras

Apr 20, 2022, SEO 636

Speaker: Alapan Mukhopadhyay, University of Michigan

Abstract: Let A be a formally unramified algebra over a field k; i.e. Ω_{A/k} = 0. When A is finite type over k, then it is well known that A is finite product of finite field extensions of k. We shall try to see whether this well known result holds without the finite type hypothesis on A. Although examples show that the well known result does not generalize to all commutative rings, under milder hypothesis on A and k, we shall prove generalization of the well known result.

Reference: Alapan Mukhopadhyay, Reducedness of formally unramified algebras over fields, arxiv.

Affine subspace concentration conditions

Mar 16, 2022, SEO 636

Speaker: Kuang Yu Wu, University of Illinois at Chicago

Abstract: We define a new notion of affine subspace concentration conditions for lattice polytopes, and prove that they hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class c_1(T_X) on Fano toric varieties.

Reference: Kuang Yu Wu, Affine subspace concentration conditions, arxiv.

Global Torelli theorem for cubic threefolds via moduli of sheaves

Apr 13, 2022, SEO 636

Speaker: Sixuan Lou

Abstract: Let X be a smooth cubic threefold, we can associate to it a principally polarized abelian variety, the intermediate Jacobian (J(X), Theta). Clemens-Griffiths and Tyurin showed the global Torelli theorem for cubic threefolds, namely two cubic threefolds are isomorphic if and only if the associated intermediate Jacobians are isomorphic as principally polarized abelian varieties. In this talk, we sketch a new proof of this result by Bayer et al, where we give an explicit parametrization of the Theta divisor in J(X) by certain moduli space of sheaves on X.

The Torelli Theorem for Algebraic Curves

Mar 30, 2022, SEO 636

Speaker: Junyan Zhao

Abstract: To an algebraic variety X, we often associate some intrinsic objects (e.g. derived categories of coherent sheaves). It is an interesting question that whether these objects encode all the informations of the variety itself? In other words, whether we can reconstruct the variety from these objects. In this talk, I will show that the Jacobian variety of a (connected) smooth projective curve C contains all of its the information. If time permits, I will talk about some similar results for smooth cubic 3-fold.

References:

1. Geometry of Algebraic Curves by ACGH

2. The Torelli Theorem for Curves by Aaron Landesman

3. Complex Abelian Varieties by C. Birkenhake

4. Mumford's book on Abelian Varieties

The tautological ring and the hyperelliptic locus

Mar 9, 2022, SEO 636

Speaker: Sixuan Lou

Abstract: The tautological ring is a canonical ring associated with M_g, the moduli space of curves of genus g. It admits many intriguing algebraic properties, giving us many insights into the geometry of M_g. I will introduce the tautological ring, its generators, and compute relations via Grothendieck-Riemann-Roch. I will also say a few words on the hyperelliptic locus and compute its class in M_3.

References:

On tautological ring:

• Harris, Morrison, Moduli of curves section

• Mumford, Towards an Enumerative Geometry of the Moduli Space of Curves

• Faber, A Conjectural Description of the Tautological Ring of the Moduli Space of Curves

https://arxiv.org/abs/math/9711218

• Vakil, The Moduli Space of Curves and Its Tautological Ring

https://www.ams.org/notices/200306/fea-vakil.pdf

On Witten's conjecture:

• NOTES ON PSI CLASSES

Applications of GRR on Jumping lines

Mar 3, 2022, SEO 636

Speaker: Junyan Zhao

Abstract: When we study a vector bundle E on projective spaces P^n, it is natural to ask the question that what can we say about the lines on P^n, restricted to which the bundle E splits unexpectedly. In this talk, we will mainly focus on the rank 2 bundle on P^2 and give an almost complete result. We will give explicit descriptions about the locus of jumping lines in some examples. Finally, I will say some words about higher dimensional cases.

Ps: to make this talk more accessible to first year students, I will give a quick review on most ingredients we need in this talk, especially cohomology and base change (I think Lawrence haven’t done this in 553) and intersection theory.

References:

• 3264 and all that Chapter 9,12,14

• Algebraic Geometry by Hartshorne Chapter 3 section 12

How to compute Grothendieck-Riemann-Roch Theorem

Feb 23, 2022, SEO 636

Speaker: Ben Gould

Abstract: We'll do some computations that demonstrate the usefulness of GRR in familiar geometric contexts.

References:

• Harris & Morrison, Moduli of curves

• Larson, A refined Brill-Noether theory over Hurwitz spaces

• Huybrechts, Fourier-Mukai transforms in algebraic geometry

Introduction to the Grothendieck-Riemann-Roch Theorem

Feb 16, 2022, SEO 636

Speaker: Anish Chedalavada

Abstract: The Grothendieck-Riemann-Roch theorem is a way to encapsulate the algebraic data of the cohomology of a vector bundle via the topological data of its characteristic classes. The fundamental insight of Grothendieck was that Riemann-Roch theorems figure as a special case of a more general manner in which characteristic classes, namely the Chern character, change with respect to proper pushforward of sheaves/algebraic cycles. In this talk I hope to motivate the definitions appearing in the theorem and prove a version for a proper map of nonsingular schemes f: X \to Y, where X is assumed to be quasi-projective.

References:

• Fulton, Intersection Theory

• Totaro, The resolution properties for schemes and stacks

F-words and how to use them

Feb 02, 2022, zoom

Speaker: Shravan Patanker

Abstract: We will define and discuss various F-words and explore their applications to rationality outside of the intended context.

References:

• F-Singularities by Polstra and Ma

• Singularities in mixed characteristic by ma and schwede

• F-rational rings have rational singularities by Karen Smith

Du Bois Singularities

Feb 02, 2022, SEO 636

Speaker: Yeqin Liu

Abstract: Du Bois singularities come naturally from Hodge theory, which is a weaker notion than rational singularities. I will talk about some connections with Hodge theory, and prove that rational singularities are Du Bois. Time permitted, I will also talk a little bit about the construction of Du Bois complex.

References:

1) Rational, Log Canonical, Du Bois Singularities On the Conjectures of Kollár and Steenbrink, Kovacs

2) Vanishing theorems on singular spaces, Steenbrink

3) The intuitive definition of Du Bois singularities, Kovacs

4) Du Bois singularities deform, Kovacs & Schwede

Rational Singularities

Jan 26, 2022, via zoom

Speaker: Junyan Zhao

Abstract: Singularities appears inevitably when we run minimal model program (MMP). In this talk, after a brief review of those somewhat technical definition of singularities in terms of discrepancy, we will introduce Cohen-Macaulay (CM) singularities and rational singularities. In particular, we will finally find that log terminal singularities are at worst rational as well as a partial converse.

References:

1) Birational Geometry of Algebraic Geometry, Kollar & Mori

2) Generalized Divisors and Reflexive Sheaves, Karl Schwede

(Isolated) Singularities in Algebraic Geometry

Jan 18, 2022, via zoom

Speaker: Ben Tighe

Abstract: The goal for the next few weeks is to understand how the MMP singularities are related to each other. This first talk will prove how these implications look in the case of isolated singularities.

References:

1) "A Survey of Log Canonical and du Bois Singularities" by Kovacs-Schwede.

2) "Singularities in the Minimal Model Program" by Kollar. Specifically chapter 3, which looks at the affine cone construction.

3) "Rational Singularities" by Kovacs. Specifically the introduction, which discusses the "inception" of rational singularities.

4) "Residues and Duality" by Hartshorne.

5) "Lectures on Vanishing Theorems" by Esnault-Viehweg.