10.14. January 
CMA Workshop
Algebraic geometry in the sciences

17. January,
CMA Geometry seminar, 13.1514, 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 Tvarieties

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 CastelnuovoSchottky
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 CalabiYau varieteter

21. March 
Modular forms, part III
Jørgen Rennemo (Oslo)
qexpansions

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^{d2}, 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<=d3 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 Lseries

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, 13.1514, 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
inclusionexclusion 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
(14.1516) 
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 basepointfree
linear series on the normalization of C, formed the corresponding map into
IP^{g1}, 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 CalabiYau varieteter

13. June 
Bank holiday

20. June 
Lunch and planning next term
