Joint work with S. Alsaody and A. Pianzola. Any group of type F_4 is obtained as the automorphism group of an Albert algebra. In the talk we show that such a group is R-trivial whenever the Albert algebra is obtained from the first Tits construction.
Let R be a noetherian (commutative) ring of Krull dimension d. A classical theorem of Forster states that a rank-n locally free R-module can be generated by n+d elements. Swan and Chase observed that this upper bound cannot be improved in general. I will discuss joint works with Zinovy Reichstein and Ben Williams where similar upper and lower bounds are obtained for R-algebras, provided that R is of finite type over an infinite field k. For example, every Azumaya R-algebra of degree n (i.e. an n-by-n matrix algebra bundle over Spec R) can be generated by floor(d/(n-1))+2 elements, and there exist degree-n Azumaya algebras over d-dimensional rings which cannot be generated by fewer than floor(d/(2n-2))+2 elements. The case d=0 recovers the folklore fact that every central simple algebra is generated by 2 elements over its center. The proof reinterprets the problem as a question on "how much versal" are certain algebraic spaces approximating the classifying stack of the automorphism scheme of the algebra in question.
Let F be a field, with absolute Galois group G. Let p be a prime. Denote by B_d the Borel subgroup of GL_d, and by U_d its unipotent radical. We consider the question of lifting a triangular Galois representation G ⟶ B_d(Z/p), to its mod p^2 analogue G ⟶ B_d( Z/p^2). It has a rich history, which we will recall. We'll then explain positive results, up to d=3, under the presence of p^2-th root of unity in F. Using an indecomposability result for divisions algebras, due to Karpenko, we'll show that the answer to the analogous question, with U_3 in place of B_3, is negative.
About 20 years ago, Reichstein obtained fundamental results on equivariant rationality for varieties with actions of finite and linear algebraic groups. New perspectives have come with the symbol invariants developed by Kontsevich, Kresch, and Tschinkel. We will discuss some implications of these ideas for rationality questions central to arithmetic geometry: complete intersections of two quadrics.
To what extent to the splitting fields of a division algebra determine its structure? In the case of maximal subfields, it turns out that this question is closely related to the famous problem "can you hear the shape of a drum?" In this talk, I'll describe some work in progress with Max Lieblich, where we consider what happens when one considers splitting fields of small transcendence degree.
We give a classification of special reductive groups over arbitrary fields that improves a theorem of M. Huruguen.
We shall discuss period/index questions for the Brauer group of function fields of hyperelliptic curves over number fields. We will relate this question to a Hasse principle for certain orthogonal Grassmannians of pencils of quadrics. We derive some consequences concerning the u-invariant in the case of genus 2 curves.
Given a smooth algebraic variety X/k, its (Bloch-Ogus-Rost) unramified cohomology H_nr(X,M) with coefficients in a cycle module M is the subgroup of M(k(X)) given by the elements whose residue at each codimension one point is zero. When X is proper, this group is a birational invariant and it being non-trivial disproves stable rationality. It was first used in this way, in the form of the unramified Brauer group, by Artin–Mumford and Saltman, and then in higher degrees by Colliot-Thélène–Ojanguren.
Given an algebraic group G/k, the cohomological invariants Inv(G,M) are the natural transformations between the functor of G-torsors over fields and the cycle module M. There are many examples dating back up to the beginning of the 20th century, but they were introduced in the present form by Serre. A result by Totaro shows that given a G-representation V such that the subset U where G acts freely has complement of codimension 2 or more, we have Inv(G,M)=H_nr(U/G,M). A few years ago, I reinterpreted the idea of cohomological invariants as invariants of the classifying stack BG, extended them to invariants of general algebraic stacks, and showed that on schemes we have Inv(X,M) = H_nr(X,M) and in fact they can be seen as the "only possible" extension of Bloch-Ogus-Rost unramified cohomology to algebraic stacks. Moreover, cohomological invariants can be used to compute Brauer groups, which I and Andrea di Lorenzo recently did for the moduli stacks of smooth Hyperelliptic curves.
Unfortunately, it's easy to see that even for smooth, projective Deligne Mumford stacks cohomological invariants are not a birational invariant. One way to see this is that while any birational map between smooth proper schemes is given, at least in char(k)=0, by a sequence of blow-ups and blow-downs, which leave cohomological invariants unchanged, for DM stacks we have to add root stacks, which can modify cohomological invariants rather drastically. I will describe recent work with Andrea Di Lorenzo in which we find a formula to describe the cohomological invariants of a root stack and use it to show that different natural compactifications of the moduli stacks of Hyperelliptic curves, while being seemingly almost identical, have vastly different cohomological invariants.
Let k be a field of characteristic zero, G be a linear algebraic k-group, and n a non-negative integer. We show that in a flat family of primitive generically free G-varieties over a base k-variety B, the points of B whose geometric fiber has essential dimension at most n form a countable union of closed subsets of B. As an application, we construct unramified non-versal G-torsors of maximal essential dimension. This is joint work with Zinovy Reichstein.
I will discuss the Hilbert scheme of d points in affine n-space, with some examples. This space has many irreducible components for n at least 3 and has been poorly understood. For n greater than d, we determine the homotopy type of the Hilbert scheme in a range of dimensions. The proof uses the homotopy theory of algebraic stacks. Many questions remain. (Joint with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson.)
Let A be a complex abelian variety. Using prismatic cohomology, we show that for all but finitely many primes p, the multiplication-by-p cover p:A\to A is p-incompressible, as conjectured by Brosnan. As an application, we obtain new p-incompressibility results for congruence covers of Shimura varieties, extending previous work of Farb-Kisin-W, Brosnan-Fakhruddin, and Fakhruddin-Saini. This is joint work with Benson Farb and Mark Kisin.
Cohomological operations in algebraic oriented cohomology theories of Levine-Morel (Steenrod operations in Chow groups; Adams operations in connective K-theory of Cai-Merkurjev; Landweber-Novikov operations and Vishik symmetric operations in algebraic cobordism) provide a useful tool to study algebraic cycles on projective homogeneous varieties G/P.
In the talk, I will show how to extend these operations to a T-equivariant setup, where T is a split maximal torus of a semisimple linear algebraic group G over a field of characteristic zero. More generally, I will show how to extend it to structure algebras of moment graphs (rings of global sections of structure sheaves on moment graphs).
I will explain a uniform algorithm that computes the usual (non-equivariant) operations for G/Ps using such extended (localized) operations and equivariant Schubert calculus techniques. This generalizes the approach suggested by Garibaldi-Petrov-Semenov for Steenrod operations. Examples include Adams operations, L.-N. operations and Vishik's Mod-p operations.