Bloch's formula
Freitag, 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
Freitag, 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
Freitag, 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
Freitag, 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.