Arbeitsgruppe Algebra -- Professor K. Altmann
Arbeitsgruppe Algebra


SFB 647  





Seminar Archiv:

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Forschungsseminar "Algebraische Geometrie" an der HU

Seminar: Algebraic Geometry Summer Semester 2010

All talks take place in room 119, Arnimallee 3.


Vortrag wegen Island-Vulkanausbruch abgesagt!
19.04.10, 16:15  Gregory G. Smith (Ontario)
Old and new perspectives on Hilbert functions
Abstract Hilbert functions are fundamental invariants in algebra geometry and commutative algebra. After recalling the basic definitions and motivating examples, we will discuss Macaulay's characterization for the collection of all Hilbert functions. We'll then contrast this with a newer viewpoint and look at potential applications.
26.04.10, 16:15  Bernd Martin (Cottbus)
03.05.10, 16:15  Jarek Wisniewski (Warschau)
Fano spaces and marked polytopes
Abstract This will be a "work in progress" report on a joint project with Klaus Altmann. I will try to explain the geometry of (some) blow ups of projective spaces (and their small modifications) from the point of view of their cones of movable divisors (and their duals).
10.05.10, 16:15  Vortrag fällt aus!
17.05.10, 16:15  Kein Seminar: Sonderseminartag am 19. Mai 2010!
19.05.10, 9:15  Orsola Tommasi (Hannover)
Diskriminantenvarietäten und der Modulraum ebener Kurven vom Grad 4 mit einer ausgezeichneten Bitangenten.
11:15  Jimmy Dillies (Utah, USA)
'Calabi-Yau varieties and K3 symmetries'
14:15  Elena Martinengo (Rom)
An overview on deformation theory: from classical techniques to infinity-groupoids.
Abstract: We will briefly recall basic tools in deformation theory via differential graded Lie algebras and show how this language allows for an immediate proof of Kodaira's principle on obstructions to deformations of complex manifolds. As further examples, we will present new results on obstructions to deformations of Hitchin pairs and on the local structure of the moduli space of locally free shaves on a given manifold. In the second part of the talk, we will reinterpret the Kodaira-Spencer approach to deformations of complex manifolds through the modern language of stacks, we will review recent results in this direction and show how this leads to infinity-groupoids as the natural and suitable language for the "local to global" approach to deformations.As illustrative examples of this, we will describe deformations of coherent sheaves and of singular varieties.
16:15  Masoud Kamgarpour (Britisch Columbia, Canada)
"Stacky abelianization of algebraic groups"
24.05.10, 16:15  Pfingsten!
24.05.10, 16:15  Marianne Merz (FU-Berlin)
Perverse Garben
Abstract: Perverse Garben haben ihren Ursprung in der Schnitthomologie. Ich beginne daher mit einer topologischen Einführung nach Goresky/MacPherson und nach der Definition perverser Garben als t-Struktur auf der derivierten Kategorie, möchte ich noch auf die wichtigsten Theoreme eingehen. Es gibt noch eine nette Anwendung, die Kalai Vermutung: g(P)>=g(F)g(P/F).
07.06.10, 16:15  Ursula Ludwig (Berlin / Freiburg)
Stratifizierte Morse-Theorie und perverse Garben
Abstract:In the nice book 'Stratified Morse Theory' (1982) Goresky and MacPherson developed a Morse theory for stratified Morse functions $f: X \ra \RRR$ on a (Whitney) stratified space $X$. Their goal was to understand better some properties of the intersection homology, a new invariant for stratified spaces they had introduced a couple of years before. In the complex case the situation is particularly nice: One gets stratified Morse inequalities relating the number of critical points of the Morse function to the intersection homology of the space. The singular points of $X$ contribute to this Morse inequalities in middle degree only and have to be counted with multiplicities. This contribution can be explained in a geometric way, but also in the more abstract language of perverse sheaves, namely in terms of the 'nearby' and 'vanishing cycles'. In this talk I will try to present the above results, which however are not directly related to my own research.
14.06.10, 16:15  NoGAGS am 17. & 18. Juni 2010
21.06.10, 16:15  Sandra Di Rocco (Stockholm)
Positivity for Toric bundles
Abstract: I will review existing results on positivity criteria for equivariant vector bundles on toric varieties and report on some conjectures and preliminary results in progress obtained with G. Smith and K. Jabbusch.
28.06.10, 16:15  Klaus Haberland (Jena)
Zur Arithmetik einiger K3-Flächen
05.07.10, 16:15  Dmitriy Pochekutov (TU-Cottbus)
On diagonals of the Laurent series of rational functions
Abstract: The diagonals of power series in several variables arise naturally in the problems of the algebraicity and $D$-finiteness of the sums of power series. We give sufficient conditions for the algebraicity of diagonals basing on the theory of multidimensional residues and topological properties of the complements to collections of complex hypersurfaces in complex analytic varieties.
12.07.10, 14:15  Maria Angelica Cueto (Berkeley)
Tropical secant graphs of monomial curves
Abstract: The first secant variety of a monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is based on a refinement of the theory of geometric tropicalization due to Hacking, Keel and Tevelev. The key step in the construction of the balanced graph involves finding a suitable compactification (a ``tropical compactification'') of the complement of a binomial arrangement in the 2-torus $(\mathbb{C}^*)^2$, whose boundary divisor has no three components intersecting at a point. Such compactification can be obtained by resolving all multiple intersections in $\mathbb{P}^2$ by blowups, and realizes the wonderful compactification of De Concini and Procesi. This is joint work with Shaowei Lin (Eprint: arXiv:1005.3364v1).
12.07.10, 16:15  Victor Batyrev (Tübingen)
A generalization of a theorem of White
Abstract: This is a joint work with Johannes Hofscheier. An n-dimensional simplex \Delta in \R^n is called empty lattice simplex if \Delta \cap\Z^n is exactly the set of vertices of \Delta . A theorem of G. K. White shows that if n=3 then any empty lattice simplex \Delta \subset\R^3 is isomorphic up to an unimodular affine linear transformation to a lattice tetrahedron whose all vertices have third coordinate 0 or 1. We prove a generalization of this theorem for an arbitrary odd dimension n=2d-1 which in some form was conjectured by Seb\H{o} and Borisov. This result implies a classification of all 2d-dimensional isolated Gorenstein cyclic quotient singularities with minimal log-discrepancy at least d
19.07.10, 16:15  Ivan V. Arzhantsev (Moskau/ Tübingen)
Equivariant compactifications of commutative unipotent groups
Abstract: The study of equivariant compactifications of a commutative unipotent group G may be regarded as an additive analogue of toric geometry. B.Hassett and Yu.Tschinkel (Int. Math. Res. Notices 20 (1999), 1211-1230) introduced a remarkable correspondence between generically transitive G-actions and finite-dimensional local algebras. We develop Hassett-Tschinkel's correspondence and calculate modality of generically transitive G-actions on projective spaces, classify actions of modality one and characterize generically transitive G-actions on projective hypersurfaces of given degree. In particular, G-actions on projective quadrics are studied. This part of the talk is based on a joint work with Elena Sharoyko. In the second part we classify flag varieties which may be realized as equivariant compactifications of G.
19.07.10, 16:15  Fedor Bogomolov (New York / Moskau)
Closed holomorphic differentials and holomorphic webs
Abstract: I will report on progress in our joint work with Bruno de Oliveira. We call a symmetric holomorphic differential $s$ on a complex manifold closed if it can be written as $\prod df_i^{n_i}$ in the local neighborhood of some point $x_o$ of the manifold. This property holds then for a complementary of a finite set of divisors and hence such a differential defines a holomorphic web--a set of local codimension one foliations on the manifold ( possibly singular). Note that any symmetric differential on surface defines a similar web. However the webs defined by closed differentials are very special and provide with restrictions on the topology of the ambient surface ( or manifold). I will discuss some results and conjectures.