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Algebraic and Arithmetic Geometry @ Bielefeld

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The Bielefeld Arithmetic Geometry Seminar 2018-2019

Welcome to the website of the Bielefeld Algebraic and Arithmetic Geometry Seminar! For more information, please contact the organizers: Michael Spiess, Jeanine Van Order, Charles Vial, and Thomas Zink.

Talks

Wednesday 5 December, 2018
15:30	U2-232	Arthur-César Le Bras (Bonn) Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve [Abstract] I will explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$.
17:00	U2-232	Andrew Kresch (Zurich) Specialization of rationality in families [Abstract] Building on work of Voisin, Colliot-Thélène and Pirutka, and Nicaise and Shinder, recent work of Kontsevich and Tschinkel establishes the specialization of rationality in smooth proper families of varieties over a field of characteristic zero and also covers the case where the special fiber is mildly singular. We describe this result and present some applications.
Wednesday 23 January, 2019
15:30	U2-232	René Mboro (Bonn) Remarks on varieties of essential $\mathrm{CH}_0$-dimension at most $2$ [Abstract] We present the notion, due to Voisin, of essential $\mathrm{CH}_0$-dimension of a variety and give two "flavors" of sufficient conditions for a variety with trivial group of $0$-cycles and essential $\mathrm{CH}_0$-dimension at most $2$ to have, in fact, essential $\mathrm{CH}_0$-dimension $0$ (i.e. to admit a decomposition of the diagonal).
17:00	U2-232	Stefan Schreieder (LMU) Stably irrational hypersurfaces of small slopes [Abstract] I will briefly recall some history of the rationality problem for hypersurfaces. I then explain how to prove that over any uncountable field of characteristic different from 2, a very general hypersurface of dimension $n>2$ and degree at least $\mathrm{log}_2(n)+2$ is not stably rational, which improves earlier results of Kollar and Totaro. I will in particular explain how the theory of quadratic forms enters the picture, which is an important ingredient in the proof.
Wednesday 30 January, 2019
15:30	U2-232	Alexander Ivanov (Bonn) On the smooth locus in infinite level Rapoport--Zink spaces [Abstract] Scholze and Weinstein constructed preperfectoid Rapoport--Zink spaces of EL-type at infinite level. Their base changes to an algebraically closed complete extension of $\mathbb{Q}_p$ are thus perfectoid. In the basic case, we will link smoothness properties of these perfectoid spaces to extra-endomorphisms of $p$-divisible groups. More precisely, we consider the special locus inside a connected component of a basic Rapoport--Zink space, corresponding to $p$-divisible groups with extra endomorphisms (e.g., in the Lubin--Tate case this is the CM-locus). Using a perfectoid version of the Jacobian criterion due to Fargues and Scholze, we prove that the complement of the special locus is a cohomologically smooth perfectoid space. A consequence of the cohomological smoothness is in particular local finiteness of the cohomology. This is a joint work with J. Weinstein.
17:00	U2-232	Shu Sasaki (Essen) Serre's conjecture about weights of mod p modular forms: old conjectures, not so old theorems, and new conjectures [Abstract] In 1987, J.P. Serre made some remarkable conjectures about weights and levels of two-dimensional modular mod p Galois representations of Gal(Q-bar/Q). They have been proved by Khare and Wintenberger (2009) building on the work of many mathematicians (Taylor and Kisin to name a few), but it has also inspired a good deal of new mathematics. One strand of research is about generalising Serre's conjectures over to totally real number fields; and Buzzard, Diamond and Jarvis made the very first attempt (2010) at doing so, while focusing exclusively on regular weights of mod p Hilbert modular forms. In my joint work with Diamond, we improve on the BDJ conjectures and formulate new conjectures about general weights of (geometric) mod p Hilbert modular forms (analogous to what Edixhoven did in 1992). I will explain what our conjectures say and demonstrate some evidence that we are on the right track.
Wednesday 29 May, 2019
15:30	U2-232	Emma Brakkee (Bonn) Pairs of K3 surfaces associated to the same cubic fourfold [Abstract] For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to K3 surfaces via their Hodge structures. For half of the d, associated K3 surfaces come in pairs. This induces an involution on the moduli space of polarized K3 surfaces of degree d. We give a geometric description of this involution. As an application, we obtain examples of Hilbert squares of K3 surfaces that are derived equivalent but not birational.
17:00	U2-232	Pieter Belmans (Bonn) Deformations of Hilbert schemes of points through derived categories [Abstract] Hilbert schemes of points on surfaces, and Hilbert squares of higher-dimensional varieties, are important and basic constructions of moduli spaces of sheaves. As such they provide a class of interesting yet tractable varieties. In a joint work with Lie Fu and Theo Raedschelders, we explain how one can (re)prove results about their deformation theory by studying their derived categories, via fully faithful functors and Hochschild cohomology, which describes both classical and noncommutative deformations.