Department of Mathematics, University of Oslo
Seminar in Algebra and Algebraic Geometry, Spring 2005

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Abstracts:


Jon Eivind Vatne, 14. January

PBW-deformations of N-Koszul algebras

An N-Koszul algebra A is a homogeneous associative algebra over a field k, where all relations are homogeneous of the same degree N, such that the minimal resolution of k as an A-module has maps alternately of degree 1 and N-1. By deforming these relations in a specific manner, we get conditions resembling the Jacobi identity for (universal enveloping algebras of) Lie algebras. The structure is then partly described by a connection which generalizes the Poincare-Birkhoff-Witt theorem; we call them PBW-deformations.
The A_\infty structure on the Ext^*_A(k,k) is particularly easy to understand. The PBW-deformations induce, and are induced by, particular deformations of this A_\infty-structure.
This talk is a report on work in progress, joint with G. Fløystad. [back to contents]

Jan O. Kleppe, 21. January

Classifying families of Gorenstein quotients of e.g. codimension 4 of a polynomial ring

Here is a pdf-file of the abstract below.

Let $B$ be a graded Cohen-Macaulay quotient of a polynomial ring $R$ with Hilbert function $H_B$. Firstly we need to recall some concepts such as ``Cohen-Macaulay'', ``Gorenstein'', ``complete intersection'' and ``licci'' and the definition and some properties of algebra cohomology. Moreover we need a Theorem of how deformations of a graded algebra $B$ are related to deformations of $A$ provided $B \rightarrow A$ is a given surjection. I recall this by some slides and I will give you a copy.
If $K_B$, the canonical module of $B$, is locally free in some open subset of $\Proj(B)$, one knows that a regular section $\sigma$ of the $B$-dual $K_B^*(s)$ ($s$ an integer) defines a Gorenstein quotient $A$ given by the exact sequence
\begin{equation} \label{K} 0 \rightarrow K_B(-s) \stackrel{\sigma}{\rightarrow} B \rightarrow A \rightarrow 0 \ \ . \end{equation}
More generally if $M_B$ is a (locally free) maximal Cohen-Macaulay module of rank $1 \leq r \leq 3$ whose top exterior power is locally a twist of $K_B$, then a regular section $\sigma$ of $M_B^*(s)$ defines a Gorenstein algebra $A$ by $(2) \ : $ $ \ A=B/\im \sigma \ $, and we have $\dim A = \dim B -r$.
Let $\GradAlg^H(R)$ be the scheme parametrizing graded quotients of $R$ with Hilbert function $H$ (this is essentially the usual Hilbert scheme if $\deg p >0$, where $p$ is the Hilbert polynomial; $p(v)=H(v)$ for $v >>0$). If $W_B \subset \GradAlg^{H_B}(R) $ is an irreducible component containing an open subset $U$ of quotients $(B)$ as above, we let $W_A \subset \GradAlg^{H_A}(R)$ be the closure of the locus of points $(A)$ constructed by \eqref{K} or (2), by varying $(B) \in U$ and $\sigma$. Under certain assumptions, notably; $s>>0$ (which we can make precise) \underline{or} $B$ in the linkage class of a complete intersection (in case \eqref{K}, the case (2) requires more), we are able to determine
(i) the dimension of $W_A$ in terms of $W_B$ and $\dim (K_B^*)_s$ and a well defined invariant $\delta$,
(ii) whether $W_A$ is an irreducible component of $\GradAlg^{H_A}(R)$ (this happens essentially when $\delta=0$), and
(iii) when $\GradAlg^{H_A}(R)$ is generically smooth along $W_A$. \\ In fact for $s>>0$ there is a well defined injective application from the set of irreducible components of $ \GradAlg^{H_B}(R) $ to the set of irreducible components of $ \GradAlg^{H_A}(R)$. In the applications we focus on algebras $A$ of codimension $4$ of $R$. [back to contents]

Martin Gulbrandsen, 21. February

Om abelske fibrasjoner av irredusible symplektiske varieteter

La X være en irredusibel symplektisk varietet av dimensjon 2n. Matsushita har vist følgende overraskende resultat: Hvis X -> Y er en ikkekonstant surjeksjon med fibre av positiv dimensjon, da er Y og alle fibrene n-dimensjonale, Y ligner svært på projektivt n-rom og en generell fiber er en abelsk varietet. Dette betyr at et studium av fibrasjoner X -> P^n kan være et skritt på veien mot en klassifikasjon av irredusible symplektiske varieteter.
Kristian Ranestad gav tidligere dette semesteret eksempler på abelske fibrasjoner for Hilbertskjemaet av n punkter på en K3. Vi skal se på eksempler på fibrasjoner X -> P^n der X er undervarieteten av Hilb^{n+1}(A) bestående av n+1 punkter på en abelsk flate A med sum lik 0. Sammen med Hilbertskjemaet av n punkter på en K3 utgjør disse 'nesten alle' kjente eksempler på irredusible symplektiske varieteter, opp til deformasjon. [back to contents]

Lars H. Halle, 28. February

Stable reduction of curves in positive characteristic

When one considers families of curves where the generic fiber is smooth, it often happens that special fibers are curves having very bad singularities.In the case where the base of the family is a regular curve, the 'stable reduction' theorem due to P.Deligne and D.Mumford states that after a finite extension in the base, one can modify the family in such a way that all fibers are 'stable' curves. In particular this means that the fibers can have no worse singularities than ordinary nodes.In characteristic 0 there is an algorithm telling you how to construct the stable reduction of a family, but no such algorithm is known in general in positive characteristic. I will discuss how one in good cases still can compute the stable reduction of families in positive characteristic. [back to contents]

Ragni Piene, 7. March

Bell-polynomer og genererende funksjoner i enumerativ geometri

For å gjette løsningen på et problem i enumerativ geometri kan man prøve å spesialisere objektene og betingelsene slik at problemet reduseres til kombinatorikk. Vi skal se eksempler på hvordan de genererende funksjoner for noen kurvetelling-problemer kan forklares på denne måten. Vi skal også se hvorfor kombinatoriske objekter som Bell-polynomer og Faa di Brunos formel for derivasjon av sammensatte funksjoner blir naturlige ingredienser i slike genererende funksjoner. [back to contents]

Olav Gravir Imenes, 14. March

Fysiske systemer beskrevet ved hjelp av moduli-rom

Prinsippet bak Laudals modell er å finne moduli-rommet for de fysiske systemene som ønskes beskrevet. Hver mulig tilstand for systemet er et punkt i moduli-rommet. Geometrien på moduli-rommet bestemmer så hvordan en tilstand går over i en annen.
Vi ser på moduli-rommet for det lokaliserte tilfellet og ser på fysiske størrelser som momentum, energi og hvilemasse, og ser deretter på Lorentz-transformasjoner. Når dette er gjort, tar vi for oss kvantemekanikk, der vi igjen ser på fysiske størrelser. [back to contents]

Bernd Sturmfels, 11. April

Tropical Geometry

Tropical geometry is the geometry of the tropical semiring (min-plus-algebra.)  Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We offer an introduction to this theory, with an emphasis on plane curves and linear spaces, and we discuss applications to phylogenetics.  [back to contents]

Bernhard Keller, 2. May

On infinitesimal deformations of derived categories

According to Kontsevich-Soibelman and Barannikov, the shifted Hochschild complex C of a differential graded algebra A over a field of characteristic 0 is the 'moduli space of A-infinity-categories'. We propose to interpret this statement to the effect that the differential graded Lie algebra C should control the deformations of the derived Morita class of A, or, in more sloppy terms, the deformations of the derived category DA. In particular, one expects a canonical bijection between the second Hochschild cohomology and the equivalence classes of infinitesimal deformations of DA. We show that such a bijection does indeed exist in many cases, notably if A itself has right bounded homology. In the general case, we obtain a bijection between the equivalence classes of Morita deformations of A and the 2-cocycle classes which act nilpotently in the graded endomorphism ring of each perfect object over A. Our proof starts from the observation that a Hochschild 2-cocycle naturally gives rise to a deformation of A in the category of curved A-infinity-algebras and that the (flat) derived category of the deformation admits a compact generator. The links of these results with Lowen-Van den Bergh's deformation theory for abelian categories remain to be elucidated. [back to contents]

Magnus D. Vigeland, 9. May

Tropisk algebraisk geometri

Tropiske varieteter er polyhedrale cellekomplekser definert av polynomer over den såkalte tropiske semikroppen (R,max,+). Fraværet av additive inverser i (R,max,+) gir opphav til en rekke interessante utfordringer når man forsøker å beskrive samspillet mellom algebra og geometri i det tropiske tilfellet. Som et spesielt eksempel vil jeg snakke om tropiske elliptiske kurver i R^3. [back to contents]

Raquel Mallavibarrena, 13. June

Scrolls over smooth curves revisited

The osculatory behavior of rational and elliptic linearly normal scrolls over curves has been studied quite extensively. In this talk I will summarize some work in progress with Antonio Lanteri where we try to extend this study to non linearly normal scrolls. I will show some interesting examples and results referred mainly to decomposable scrolls. [back to contents]