Projectivity of Rigid Group Actions on Complex Tori
Friday, 19.10.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we shall discuss a result already obtained by Torsten Ekedahl around 1999, stating that every complex torus \(T\) admitting a rigid group action of a finite group \(G\) is in fact projective, i.e., an Abelian variety. Firstly, we shall explain the notion of “deformations of the pair \((T,G)\)“; afterwards the proof of Ekedahl’s Theorem will be outlined and the projectivity of \(T\) will be shown explicitly. If time allows, applications of Ekedahl’s result will be explained towards the end of the seminar talk. This is (partly) joint work with Fabrizio Catanese.\n\n
Finiteness of perfect torsion points of an abelian variety
Friday, 26.10.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
I will report on a joint work with Emiliano Ambrosi. Let k be a field\nthat\nis finitely generated over the algebraic closure of a finite field. As\na\nconsequence of the theorem of Lang-Néron, for every abelian variety\nover k\nwhich does not contain any isotrivial abelian variety, the group of\nk-rational torsion points is finite. We show that if k^perf is a\nperfect\nclosure of k, the group of k^perf-rational torsion points is finite as\nwell. This gives a positive answer to a question asked by Hélène\nEsnault in\n2011. To prove the theorem we translate the problem to a certain\nquestion\non morphisms of F-isocrystals. Subsequently, we handle the question\nstudying the monodromy groups of the F-isocrystals involved. We can\nprove\nthat a certain monodromy group is "big" via an argument with Frobenius\ntori. Then class field theory and some considerations on the slopes\nconclude the proof. As an additional outcome of our work we prove a\nweak\n(weak) semi-simplicity statement for p-adic representations coming\nfrom\npure overconvergent F-isocrystals.\n
Bloch's formula
Friday, 2.11.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we discuss Bloch's formula for smooth and singular schemes. The formula relates Chow group of cycles on a scheme with the cohomology of K-sheaves or K^M-sheaves, where K and K^M stand for K-theory and Milnor K-theory, respectively. In smooth case, the formula is a corollary to the Gersten resolution. As Gersten resolution for these sheaves is not available on singular schemes, in a joint work with Prof. Amalendu Krishna, we use Cousin complex to study the Bloch's map. \n\nWe begin the talk by recalling the definition of Chow groups and Milnor K-groups and briefly discuss the formula for smooth schemes. In the case of singular schemes, we use Cousin complex to define Bloch's map. We then prove the formula for affine schemes over algebraically closed fields and for regular in codimension one projective schemes over algebraically closed fields. At last, Bloch's formula with modulus will be discussed. \n\n
Geometry of intersections of some secant varieties to algebraic curves
Friday, 9.11.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. In this talk we consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We will describe some interesting cases with unexpected transversality properties and, if time permits, explain a general method to verify when this intersection is empty.
Two polarized K3 surfaces associated to the same cubic fourfold
Friday, 23.11.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
For infinitely many d, Hassett showed that special cubic fourfolds of\ndiscriminant d are related to polarized K3 surfaces of degree d via\ntheir Hodge structures. For half of the d, a generic special cubic has\nnot one but two different associated K3 surfaces. This induces an\ninvolution on the moduli space of polarized K3 surfaces of degree d. We\ngive a geometric description of this involution. As an application, we\nobtain examples of Hilbert schemes of two points on K3 surfaces that are\nderived equivalent but not birational.
Chow schemes in mixed characteristic
Friday, 30.11.18, 10:30-11:30, Hörsaal, Otto-Krayer-Haus
Spaces parametrizing positive algebraic cycles have been in use in algebraic geometry for a long time.\nHowever in positive and mixed characteristic we do not know to which extent these spaces can be understood in terms of moduli problems. Some progress has been made however:\nin '96 Suslin and Voevodsky introduced a presheaf of effective relative zero cycles on the category of normal varieties and proved that it is isomorphic to the presheaf represented by infinite symmetric powers (after localization by the characteristic of the field when it is positive). The aim of this talk is to explain how Suslin and Voevodsky's theorem\ncan be generalized to schemes of mixed characteristic and also to higher dimensional cycles. We intend the talk to be understandable for algebraic geometers of various backgrounds thus we start by recalling the definition of a relative cycle and give an insightful example as well as introduce other useful notions such as Voevodsky's h-topology.\nAfter stating our theorem we briefly explain the strategy behind its proof and give a relatively detailed proof of one of the key components.
Differential transcendence of special functions
Friday, 14.12.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
One of the goal of the difference Galois theory is to\nunderstand the algebraic relations between solutions of a linear\nfunctional equation. Recently, Hardouin and Singer developed a Galois\ntheory that aims at understanding what are the algebraic and\ndifferential relations among solution of such equations. In this talk we\nare going to see recent results ensuring that in many situations, such\nsolutions satisfy no algebraic differential relations.
p-adic variations of automorphic sheaves
Friday, 11.1.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Elliptic modular forms are a special kind of functions on the Poincare' upper half space and have played an increasingly important role in modern Number Theory. Starting with the works of J.P. Serre and N. Katz more than 30 years ago, it was discovered that, given a prime number p, such modular forms have also a p-adic nature and, especially, live in p-adic families. This phenomenon is the counterpart of the theory fo p-adic deformations of Galois representations and has become a basic tool for number theorists. I will present joint work with A. Iovita and V. Pilloni providing a geometric explanation of this, purely p-adic, phenomenon.\n
Computing classes of admissible covers
Friday, 18.1.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Let Adm(g,h,G) be the space of degree admissible G covers C → D of a genus h curve D by genus g curves C. There is a natural map f : Adm(g,h,G) → Mgnbar into the moduli space of stable curves taking the source curve of an admissible cover and forgetting everything else. When the class [f(Adm(g,h,G))] is tautological we can try to express this class in terms of a known basis for the tautological ring of Mgnbar. We will discuss several strategies for making these computations and give a number of examples.\n
Rigidity for equivariant K-theory
Friday, 25.1.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
This talk is a report on joint work with Jeremiah Heller and Paul Arne Østvær. The Gabber-Gillet-Thomason rigidity theorem asserts that the natural map from a henselian local ring to its residue field induces an isomorphism on algebraic K-theory with finite coefficients (coprime to the exponential characteristic). We establish a version of this rigidity theorem in the setting of homotopy invariant equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic K-theory and presheaves with equivariant transfers.
Rigid rational curves in positive characteristic
Friday, 1.2.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Rational curves are central to higher-dimensional algebraic geometry. If a rational curve “moves” on a variety, then the variety is uniruled and in characteristic zero, this implies that the variety has negative Kodaira dimension. Over fields of positive characteristic, varieties can be inseparably uniruled without having negative Kodaira dimension. However, I will show in my talk that in the case that a rational curve moves on a surface of non-negative Kodaira dimension, then this rational curve must be “very singular”. In higher dimensions, there is a similar result that is more complicated to state. I will also give examples that show the results are optimal. This is joint work with Kazuhiro Ito and Tetsushi Ito.
Motives on general base "spaces"
Friday, 22.3.19, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk I will begin with an introduction to motives. I will then\ndefine a category of motives on very general base spaces, so-called\nprestacks. This framework allows us to consider motives on, say, an\ninfinite-dimensional affine space, and also equivariant motives. If time\npermits I will sketch an application of this formalism to a motivic\nSatake equivalence, a cornerstone in the Langlands program. My intention\nis to keep the talk as non-technical as possible. This is joint work\nwith Timo Richarz.