Seminario di geometria algebrica e aritmetica di Pisa |
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This talk is based on joint work with Yves André dedicated to the study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the fpqc topology and the canonical topology (i.e., the finest topology for which all representable presheaves are sheaves).
After a short introduction to Grothendieck topologies, we shall focus on the category of affine schemes and present our reconstruction of Olivier's proof that the canonical topology coincides with the effective descent topology: covering maps are given by universally injective ring maps. Grothendieck proved that the fpqc topology is coarser than the canonical topology. Actually, it is strictly coarser: we propose new examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology).
"Splinters" are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. Time permitting, we present in geometric terms the state of the art about them.
I will start by presenting some applications of the minimal model program to the construction of symplectic compactifications of certain Lagrangian fibrations (integrable systems) of geometric origin. One such case is that of the intermediate Jacobian fibration associated to a cubic fourfold (Donagi-Markman integrable system). In the second part of the talk, I will discuss aspects of the birational geometry of this fibration (e.g. the Mordell-Weil group).
In this talk we study certain moduli spaces of semistable objects in the Kuznetsov component of a cubic fourfold. We show that they admit a symplectic resolution M˜ which is a smooth projective hyperkaehler manifold deformation equivalent to the 10-dimensional example constructed by O'Grady. As a first application, we construct a birational model of M˜ which is a compactification of the twisted intermediate Jacobian of the cubic fourfold. Secondly, we show that M˜ is the MRC quotient of the main component of the Hilbert scheme of elliptic quintic curves in the cubic fourfold, as conjectured by Castravet. This is a joint work with Chunyi Li and Xiaolei Zhao.
I will describe a class of stack-theoretic modifications, and sketch how they are applied in resolution of singularities. This is a concrete exposition of work with Temkin and Wlodarczyk and of work of Quek.
When studying the zeta functions of algebraic varieties over finite fields of characteristic p, there are essentially two approaches to put these in the context of a Weil cohomology theory. One approach is the familiar one using etale cohomology; in this theory, the cohomology groups are vector spaces over an l-adic field where l is distinct from p. The other is the one derived from Dwork's proof of rationality of zeta functions and incorporating Berthelot's work on crystalline cohomology.
We first describe the locally constant coefficient objects in these two
approaches; in the etale approach these are representations of certain
fundamental groups (lisse l-adic sheaves), while in the crystalline
approach these are certain vector bundles with connection
(overconvergent F-isocrystals). We then assert a theorem that says that
these objects do not occur in isolation: on a smooth variety over a
finite field, any object in one category admits corresponding objects in
all of the other categories which carry the same arithmetic information.
This builds on the Langlands correspondence for GL_n (Drinfeld, L.
Lafforgue, Abe), which shows that on a curve, coefficient objects always
have "geometric origins". It also incorporates results of Drinfeld,
Deligne, Abe, Esnault, and the speaker, which together work around the
fact that geometric origins on higher-dimensional varieties seem to be
quite hard to establish.