Algebraic Geometry Seminar

Kawamata-Viehweg vanishing is very useful in birational geometry. I will generalize Kawamata-Viehweg vanishing and Nadel vanishing for multiplier ideals to Saito's Hodge modules. Meanwhile, I will construct multiplier subsheaves for Hodge modules generalizing multiplier ideals. As a application, I will prove a Fujita type freeness result for Hodge modules which generalizing a result of Kamawata for higher direct images of dualizing sheaves. If time permits, I will also explain how the multiplier subsheaves related to the V-filtration of holonomic D-modules.

Give a smooth variety Y with an action of a finite group G, and given a semiorthogonal decomposition of the derived category D[Y/G] of G-equivariant coherent sheaves on Y into subcategories that are equivalent to derived categories of smooth varieties, we construct a semiorthogonal decomposition for a smooth G-invariant divisor on Y. Combining this with some known semiorthogonal decompositions when G is a complex reflection group acting on a vector space, we construct semiorthogonal decompositions of some G-equivariant derived categories of projective hypersurfaces.

Inspired by the recent solution of the direct summand conjecture of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed characteristic. As an application, we obtain a uniform bound on the growth of symbolic powers in regular rings of mixed characteristic analogous to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal characteristic. This is joint work with Karl Schwede.

Supersingular K3 surfaces are a remarkable class of varieties that exhibit many unique features of positive characteristic algebraic geometry. We will give a new construction of certain one-parameter families of (twisted) supersingular K3 surfaces, which we call supersingular twistor lines. This will entail a detailed examination of the Brauer group, together with techniques coming from the derived category and Fourier--Mukai equivalences. We will show how these families are a powerful tool in the study of supersingular K3 surfaces, yielding relatively short proofs of Ogus' crystalline Torelli theorem, Rudakov--Shafarevich's results on degenerations of supersingular K3 surfaces, and Shioda's conjecture on unirationality. As a byproduct, we extend these results to a wider range of characteristics than was previously known.

Azumaya algebras are (etale) twisted forms of matrix rings. These objects are of great utility because they give rise to Brauer classes. Fifty years ago, Grothendieck asked whether every cohomological Brauer class has a corresponding Azumaya algebra. This question is still open even for smooth separated threefolds over the complex numbers! One says a scheme (or Algebraic stack) X satisfies the resolution property if every coherent sheaf is the quotient of a vector bundle. The work of Totaro and Gross explains that this property holds iff X admits a very special quotient stack presentation. However, whether or not separated Algebraic stacks have this property remains a difficult question. The goal of our talk will be to explain (1) Why these two questions are deeply intertwined, (2) New results regarding the existence of Azumaya Algebras and (3) How we can use results as in (2) to show large classes of algebraic stacks satisfy the resolution property.

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.

Mori and Mukai conjectured that projective space should be the only n-dimensional Fano variety whose anti-canonical bundle has degree at least n + 1 along every curve. While this conjecture has been proved in characteristic zero, it remains open in positive characteristic. We will present some progress in this direction by giving another characterization of projective space using Seshadri constants and the Frobenius morphism. The key ingredient is a positive-characteristic analogue of Demailly’s criterion for separation of higher-order jets by adjoint bundles.

Let L be a line bundle on a projective variety X. We use valuations to measure the singularities of the linear system |mL| as m goes to infinity. Specifically, we consider the global log canonical threshold and stability thresholds of L. The stability threshold generalizes an invariant recently introduced by Kento Fujita and Yuji Odaka. When X is a Fano variety, we show that the stability threshold detects the K-(semi)stability of X. This talk is based on joint work with Mattias Jonsson.

A Kodaira fibration is a non-isotrivial fibration f: S --> B from a smooth algebraic surface S to a smooth algebraic curve B such that all the fibers are smooth algebraic curves. Such fibrations thus arise as complete curves inside the moduli space M_g of genus g algebraic curves. We investigate the possible connected monodromy groups of a Kodaira fibration in the case g=3 and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of M_3.

The LCT (log canonical thresholds) is an invariant which measures the complexity of singularities. A conjecture due to Shokurov predicts that LCT for a single divisor satisfy the ACC (ascending chain condition), and it was proven by Hacon-McKernan-Xu. In this talk, I will sketch the proof of the ACC of LCT polytopes (for multiple divisors). This is a joint work with Zhan Li and Lu Qi.

The degree of irrationality of an n-dimensional algebraic variety is the minimal degree of a dominant rational map from the variety to n-dimensional projective space. While of a classical flavor, this invariant has not received much attention in dimension n>1 until recently. In this talk we will explain some new results on the computation of this invariant for hypersurfaces in certain Fano varieties. We emphasize the connection between low-degree maps from these hypersurfaces to projective space and the geometry of low degree curves inside the Fano varieties. These results extend recent work of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery. This is joint work with Brooke Ullery.