Last modified: Mon Aug 15 12:50:07 CEST 2011 or later by ingerbo[at]math.uio.no.

### The Algebra and Algebraic geometry group Department of Mathematics, University of Oslo

General information about our mailing list algebra-seminar[at]math.uio.no can be found here.
Seminar history: Spring 2003, Autumn 2003, Spring 2004, Autumn 2004, Spring 2005, Autumn 2005, Spring 2006, Autumn 2006, Spring 2007, Autumn 2007, Spring 2008, Autumn 2008, Spring 2009, Autumn 2009, Spring 2010, Autumn 2010.
Fridays 14.00 for 14.30, pausearealet 7. etasje, Felleskollokviet (Department Colloquium). Contact: Bjørn Jahren (bjoernj).

 Seminar in Algebra and Algebraic geometry Time: Mondays 14.15 (lunch at 12.00, 6. floor) Venue: Seminar room B63, Niels Henrik Abel building Organiser: Arne B. Sletsjøe (arnebs)

## NB!     Next seminar:

10.-14. January CMA Workshop
Algebraic geometry in the sciences
17. January,
CMA Geometry seminar,
13.15-14, B1036
Georg Muntingh (Oslo, CMA)
Generalized principal lattices in space
Abstract:
In multivariate polynomial interpolation theory, the properties of polynomial interpolants depend very much on the configuration of the interpolation points in space. An important class is made up by the generalized principal lattices, which form a corner stone in the classification of the meshes with simple Lagrange formula and can be viewed as a generalization of the triangular meshes.
While generalized principal lattices are defined by an abstract combinatorial definition, all generalized principal lattices in the projective plane arise from a real cubic curve in the dual projective plane. As all such curves are of arithmetic genus 1, one can ask the question: Which space curves of arithmetic genus 1 and degree 4 give rise to generalized principal lattices in dual projective space?
In this talk we show how generalized principal lattices arise naturally from the notion of a triangular mesh and try to give an answer to this question.
24. January Lunch
31. January Seminar series on Modular forms, part I
Lars Halvard Halle (Oslo)
Modular group and modular curve
7. February Peter Arndt (Oslo)
The field with one element
Abstract:
The 'field with one element' is the name given to a hypothetical mathematical entity which is supposed to lie behind several limit phenomena of geometry and algebra over finite fields and is hoped to provide an explanation of the analogy between function fields and number fields. In this talk we will first present some of the observations which gave rise to the idea of such an entity, then make a quick survey of several proposed definitions of geometry over F_1, and finally take a glimpse into proposed compactifications of Spec Z which can be constructed in some of these geometries.
14. February Peter Arndt (Oslo)
The field with one element, part 2
21. February Robert Volmert (Freie Uni. Berlin)
Deformations of affine T-varieties
28. February Modular forms, part II
Arne B. Sletsjøe (Oslo)
Modular functions/Uniformization/Fields of moduli
7. March Martin Gulbrandsen (Stord/Haugesund)
Effective Torelli via the punctual Hilbert scheme
Abstract:
I will explain a construction that recovers a curve from its Jacobian X as a certain locus in the Hilbert scheme of finite (nonreduced) subschemes in X, defined in terms of the canonical polarization on X. The construction is as explicit as the Hilbert scheme is. This is an application of the Castelnuovo-Schottky theorem I talked about at the national algebra meeting in Bergen, about which I will say a bit more. A prerequisite for the audience is to have forgotten the Bergen talk. This is joint work with M. Lahoz.
14. March Matthias Schütt (Hannover)
Oversiktsforedrag over modularitet til Calabi-Yau varieteter
21. March Modular forms, part III
Jørgen Rennemo (Oslo)
q-expansions
28. March Andrea Hofmann (Oslo)
Idealet, syzygier og trisekanter til kurver av genus 2
Sammendrag:
Gitt en kurve C av genus 2 og grad d>=6 embedded i P^{d-2}, så ønsker man for eksempel å gi en god geometrisk beskrivelse av kurven og en i en viss forstand enkel beskrivelse av idealet I_C.
Siden enhver g^1_k på C med k<=d-3 gir opphav til en rasjonal normal skrue av dim. k som inneholder C, så ønsker man å studere problemstillingen om man får en dekomposisjon av idealet I_C i idealer til rasjonale normale skruer definert vha g^1_k'er på C.
Vårt resultat er at I_C=I_S+I_V, der S er den entydige g^1_2(C)-skruen, og V er en g^1_3(C)-skrue som ikke inneholder S.
Vi skisserer beviset for dette og går så over til et naturlig oppfølgerspørsmål: Gitt en kurve C, finnes det rasjonale normale skruer (som inneholder C) slik at de første syzygiene til I_C er generert av de første syzygiene til idealene til disse rasjonale normale skruene? Vi gir eksempler på kurver C av grad 7 der dette er tilfellet.
Til slutt finner vi graden til den tredje sekantvarieteten til C ved å identifisere denne varieteten som union av alle g^1_3(C)-skruer.
4. April Modular forms, part IV
Nikolay Qviller (Oslo)
Hecke operators and L-series
11. April Johannes Nicaise (K.U. Leuven)
Rational fixed points for finite group actions, and a question of Serre
Abstract:
Serre asked if the action of a finite $\ell$-group on an affine space over a field of characteristic different from $\ell$ always admits a rational fixed point. We will give a brief survey on known results and techniques, and we will show how one can use Loeser and Sebag's motivic Serre invariant to answer the question affirmatively if the base field is a henselian discretely valued field of characteristic zero with algebraically closed residue field of characteristic different from $\ell$. This is joint work with Hélène Esnault.
18. April Easter break
25. April Easter break
2. May Modular forms, part V
Lars Halvard Halle (Oslo)
Modularitet av elliptiske kurver
9. May,
CMA Geometry seminar,
13.15-14, B1036
Nikolay Qviller (Oslo)
Fundamental polynomials of nodal curve counting
Abstract:
Nodal curves in a fixed complete linear system |L| on a projective surface S are enumerated by Bell polynomials in certain universal, linear combinations of the four Chern numbers of (S,L). These linear polynomials can be interpreted through the contribution of certain diagonals to certain intersection products, but have proven difficult to describe explicitly. It turns out that part of the reason is the failure of the Segre class of a closed subscheme to satisfy an inclusion-exclusion principle. However, as was pointed out by Aluffi, this failure can be corrected by slightly modifying the definition of the Segre class. After recalling some essential points in the enumerative geometry of nodal curves, I will discuss the application of Aluffi's ideas to this particular problem.
9. May
NB! Room B1036
(14.15-16)
CMA Guest Lecture
Steven Kleiman (MIT)
The Canonical Model of a Singular Curve
Abstract:
Given an arbitrary complete integral curve C of arithmetic genus g at least 2, Rosenlicht (1952) introduced its canonical sheaf \omega and its canonical model C': he proved that H0(\omega) defines a base-point-free linear series on the normalization of C, formed the corresponding map into IP^{g-1}, and took C' to be its image. Rosenlicht proved, among other results, that C and C' are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C' is equal to the blowup of C with respect to \omega.
This talk will report on some recent joint work of Renato Vidal Martins and the speaker, which provides refined statements and modern proofs of Rosenlicht's results, plus a determination of just when C' is rational normal, arithmetically normal, projectively normal, and linearly normal.
16. May Lunch
23. May The Holmboe prize ceremony and symposium
30. May Lunch
6. June Modular forms, part VI
Lars Halvard Halle (Oslo)
Modularitet av Calabi-Yau varieteter
13. June Bank holiday
20. June Lunch and planning next term