Department of Mathematics, University of Oslo
Seminar in Algebra and Algebraic Geometry, Autumn 2006

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Abstracts:


Kristian Ranestad, 25. September

The algebraic degree of semidefinite programming

Gitt et generelt lineært m-dimensionalt rom av symmetriske (n x n)-matriser. Hva er graden til den duale varieteten til undervarieteten av matriser av rang høyst r (< n), og hva har denne graden med semidefinit programmering å gjøre? Jeg vil rapportere fra pågående arbeid med Jiawang Nie og Bernd Sturmfels. [back to contents]

Andrea Hofmann, 2. October

Kanoniske kurver i rasjonale normale skruer

Gitt en abstrakt ikke-singulær ikke-hyperelliptisk kurve C av genus g kan vi embedde den inn i P^{g-1} ved hjelp av det kanoniske lineære systemet. Kurven vil da ligge i sin gonalitetsskrue T, og vi kan beregne resolusjonen av O_C som O_T-modul, og deretter resolusjonen av O_C som O_{P^{g-1}}-modul. For trigonale og tetragonale kurver er resolusjonen vi får minimal, slik at vi får Betti-tallene direkte fra resolusjonen. For høyere gonaliteter er resolusjonen ikke nødvendigvis minimal, slik at vi ikke kan bestemme Betti-tallene ut fra resolusjonen, men vi kommer med en formodning om hvordan Betti- tallene ser ut.
Hvis jeg får tid, skal jeg også snakke litt om tilfellene der vi ikke jobber med gonalitetsskruen men med Cliffordskruen for kurver av Clifforddimensjon 2 og 3.
Dette blir et foredrag om masteroppgaven min. [back to contents]

Jan Kleppe, 16. October

The Hilbert scheme of codimension 2 subschemes (mainly in P^3) and related moduli spaces of reflexive sheaves

Let $H(d,g)$ be the Hilbert scheme of space curves of degree $d$ and arithmetic genus $g$ and let $\bf H_C$ be the local Hilbert functor of flat deformations $C_S \subset {\bf P^3} \times S$, S a local artinian $k$-algebra, at a fixed curve $C$. An effective method of studying the local ring of $H(d,g)$ at $(C)$ with respect to e.g. smoothness, dimension and irreducibility, is to look at other local deformation functors $\bf D$ over $\bf H_C$, $\bf D \rightarrow \bf H_C$, which allows a surjective tangent map $t_{\bf D} \rightarrow t_{\bf H_C} = H^0({\it N}_C)$ and a corresponding injective map of obstruction spaces. We consider several such deformation functors $\bf D$ which determine $H(d,g)$ locally under various assumptions. In particular we look to the functor {\bf (1)} of Hilbert function stratum, GradAlg(H), and {\bf (2)} of deformations of a pair $(C,\xi)$ where $\xi$ is an extension in the Serre-correspondence \vskip 5 pt \centerline {$ \xi \ \ ; \ \ 0 \rightarrow O_{\bf P^3} \rightarrow F \rightarrow I_C(c_1) \rightarrow 0 \ .$} \vskip 5 pt The former leads to explicit criteria of obstructedness/unobstructedness and related dimension formulas for $H(d,g)$. The latter allows us to prove several results for the moduli space of reflexive sheaves on ${\bf P^3}$, e.g. we make explicit necessary and sufficient conditions for the unobstructedness of a reflexive arithmetically Buchsbaum sheaf of diameter one. [back to contents]

Arne B. Sletsjøe, 23. October

Deformasjoner av ekstensjoner av moduler

I ikke-kommutativ algebraisk geometri studerer vi endelig- dimensjonale representasjoner av ikke-kommutative algebraer. Disse representasjonene har en interessant egenskap som vi ikke observerer kommutativt, nemlig at det finnes ikke-splitt ekstensjoner av ikke- isomorfe moduler. Ekstensjoner av denne typen deformerer enten til simple moduler eller til andre ekstensjoner. I studiet av slike deformasjoner er homologisk algebra et godt verktøy. Vi skal se litt nærmere på dette verktøyet og bruke det til å prøve å forstå den lokale strukturen til det formelle modul-rommet i et punkt svarende til en ekstensjon. [back to contents]

HiBu, Kongsberg, 7. November

Algebraseminar at HiBu

Eivind Eriksen
Title: Noncommutative deformations of differential structures
Abstract: Let k be an algebraically closed field of characteristic 0, and let X be a algebraic variety over k. We consider (1) the category of quasi-coherent D_X-modules and (2) the category of integrable g-connections for a fixed Lie algebroid g of X, and explain how to compute noncommutative deformations in these categories.
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Trond Gustavsen
Title: Connections on modules over singularities
Abstract: Let A be a commutative k-algebra, where k is an algebraically closed field of characteristic 0, and let M be an A-module. We consider the following question: Under what conditions on A and M is it possible to find a connection on M?
I will report on joint work with Eivind Eriksen and Runar Ile.
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Daniel Larsson
Quasi-Lie algebras and Quasi-Deformations
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Johannes Kleppe
Multilinear maps
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Andrew Dancer, 13. November

Adding branes to hyperkahler manifolds

We investigate an analogue for hyperkahler manifolds of the symplectic cut construction. In low dimensions this has an interpretation in terms of adding branes to hyperkahler manifolds. [back to contents]

George Hitching, 20. November

Tangent cones to generalised theta divisors

Let X be a curve of genus at least 3, and U the moduli space of semistable vector bundles of rank r and slope g-1 over X. The variety U has a natural divisor whose support consists of bundles with nonzero sections. We give a geometric description of the tangent cone to this divisor at a stable point, which generalises the Riemann-Kempf singularity theorem for line bundles of degree g-1 over X. As a corollary, we give a generalisation of the geometric Riemann-Roch theorem to bundles of arbitrary rank. [back to contents]

Oleg Viro, 27. November

Tropical geometry and patchworking

Tropical geometry studies tropical varieties which are piecewise linear objects closely related to algebraic varieties. Tropical varieties can be obtained as a limit, a sort of dequantization of algebraic varieties. In many problems tropical varieties can be used instead of algebraic varieties. One can construct algebraic varieties as perturbations of tropical ones. Tropical geometry emerged during the last ten years from different sources. One of its sources was patchworking of real algebraic hypersurfaces, the simplest version of which can be considered as the constructing of a real algebraic variety by perturbation (or quantization) of a tropical one. [back to contents]

Ragni Piene, 4. December

Inflectional loci of scrolls

Let X be a projective scroll, i.e., a variety ruled by linear spaces over a smooth curve. We show that certain natural sheaves associated with the jet bundles of X are locally free, and we use these sheaves to express the class of the inflectional locus of X. In particular we obtain explicit formulas counting the number of flexes when there are only finitely many.
This is joint work with Antonio Lanteri and Raquel Mallavibarrena. [back to contents]

Lutefisk-seminaret, 12. December

Sandra Di Rocco

Title: 'Q-factorial toric varieties covered by lines, geometrical and combinatorial applications'
Abstract:
A toric variety is a normal algebraic variety endowed with an algebraic action of a torus and containing a dense open orbit. Toric varieties represent an important class of examples in algebraic geometry as the torus action induces a combinatorial structure on the variety. A characterization of Q-factorial toric varieties covered by lines will be presented. Such varieties can be described in terms of an elementary toric fibration, i.e. a surjective equivariant morphism onto a Q-factorial toric variety whose fiber has Picard number one. This result leads to the classification of projective Q-factorial toric varieties with positive dual defect.
If time permits, a combinatorial application leading to the characterization of the discriminant of certain configurations of lattice points will be presented. This is joint work with C. Casagrande.
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Dan Laksov

Tittel: 'Dividerte potensalgebraer og invarianter i flere variable'
Sammendrag:
Vi observerer at dividerte potensalgebraer kan brukes til å beskrive invarianter av ringer i flere familier av variable, under den symmetriske gruppen, når den permuterer de variable simultant. Dette gir et nytt synspunkt på noen klassiske resultater.
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