Offisiell side


Seminar in Algebra and Algebraic Geometry
Spring 2010


Welcome to the seminarpage

    The Algebra and Algebraic Geometry group

Department of Mathematics, University of Bergen



Seminar history: Spring 2008, Autumn 2008, Spring 2009, Autumn 2009 .


Time:   Friday 14:15-16:00
Venue:   Room 534, Carl L. Godskes hus,
  Johannes Brunsgate12
Organiser:   Andreas Leopold Knutsen
Friday, February 5th Gunnar Fløystad (UiB)
Betti diagrammer til Zn-graderte artinske moduler over polynomringen k[x1, ..., xn].
Abstract:
Tematisk hører foredraget inn under gjennombruddet som er kommet de senere årene innen studiet av Betti diagrammer til graderte moduler over polynomringer, som følge av formodningene til Boij og Soederberg.

Vi studerer det lineære rommet og den positive kjeglen generert av Betti diagrammer til Zn-graderte artinske moduler over polynomringen i n variable, som blir rene (pure) med gitte grader d0, ..., dn når man tar totale grader.

Vedrørende det lineære rommet viser vi at dette har en naturlig basis (entydig utifra visse kriterier) bestående av alle tvister til Betti-diagrammet til den ekvivariante rene resolusjonen som ble konstruert for et par år siden.

Vedrørende den positive kjelgen klassifiserer vi de ekstremale strålene når vi har en polynomring i to variable.

Friday, February 5th Gunnar Fløystad (UiB)
Betti diagrammer til Zn-graderte artinske moduler over polynomringen k[x1, ..., xn]. DEL II
Abstract:
Dette er en fortsettelse fra forrige uke.
Friday, February 12th Nils Henry Rasmussen (UiB)
Brill-Noether-teori og Green-formodningen for kurver på K3-flate
Abstract:
Vi vil beregne dimensjonen av pensler på glatte kurver på K3-flater S ved studere Lazarsfeld-Mukai-vektorbuntene. Disse er rang 2-vektorbunter på S som følger med hver gang vi betrakter en kurve C og en pensel A. Ved betrakte ulike egenskaper disse vektorbuntene må ha, kan vi finne en ølvre skranke for dimensjonen av pensler på en generell C ved å finne en skranke for dimensjonen av Lazarsfeld-Mukai-vektorbunter. En konsekvens av skranken vi finner, er at Green-formodningen gjelder for alle kurver på K3-flater. Dette føllger fra et resultat av Aprodu i 2005.
Friday, February 19th Alessandra Sarti (Université Poitiers, France)
Automorphisms of holomorphic symplectic manifolds and Enriques varieties
Abstract:
Holomorphic symplectic manifolds are special Kähler manifolds with zero first Chern class. Together with Calabi-Yau manifolds they can be considered as a higher dimensional analogous of K3 surfaces. In the first part of the talk I will recall basic definitions and properties, then I will study their automorphism group. I will present recent results and introduce the notion of Enriques variety, which generalize that of Enriques surface. The latter are exactly the quotients of K3 surfaces by fixed point free involutions and we give a similar definition in higher dimension. In the last part of the talk I will construct explicit examples of Enriques varieties. The results I will present are obtained in several papers in collaboration with Samuel Boissière and Marc Nieper-Wisskirchen.
Friday, February 26th Andrea Hofmann (UiO)
Title: The ideal and trisecants of a curve of genus 2
Abstract:
Given a curve C of genus 2, it is known that there exists exactly one linear system g12 on C and that the family of g13's on C is two-dimensional, isomorphic to the Jacobian variety of C. We can embed C with a linear system of degree d>=6 in Pd-2. The g12(C) gives rise to a rational normal scroll S of dim. 2 and each g13(C) gives rise to a three-dimensional rational normal scroll V. All these scrolls contain the curve by construction, so the ideals IS and IV are contained in the ideal IC, and a natural problem for discussion is now given by the question if IS and IV generate the whole ideal of C whenever V is a g13(C)-scroll that does not contain S. The answer to this question is positive in several cases for d and we sketch the proof for d=7. Also we will look at the analogous question for the higher syzygies of IC, i.e. are the higher syzygies of IC generated by the higher syzygies of IS and the higher syzygies of all IV? Here we focus on the first syzygies of IC and the case when the degree of the curve is equal to 7. The third secant variety of a curve C, Sec3(C), is the (Zariski-) closure of the union of all planes that are spanned by three points on C. It can be shown that for C of degree d>=8 the degree of Sec3(C) is equal to the number of trisecant lines to a curve of genus g and degree d in P4. Here the genus of the curve is equal to g=2, and one can show that the union of all g13(C)-scrolls is isomorphic to Sec3(C). We use this result to show (once more) Berzolari's formula (for g=2), which gives the number of trisecant lines to a curve of genus g and degree d (d>=6) in P4.
Friday, March 5th Martin G. Gulbrandsen (Høyskolen Stord/Haugesund)
Title: Donaldson-Thomas invariants with a view toward abelian threefolds
Abstract:
Deformation theory suggests that moduli spaces for stable coherent sheaves on Calabi-Yau threefolds are expected to be finite. In reality this is seldom the case. Still the number of points a moduli space should have had, if it were finite, can be defined, and these numbers are the Donaldson-Thomas invariants of the CY threefold. The talk will be expository. I will outline the construction of DT invariants via virtual fundamental classes, together with Behrend's important insight, that they can be expressed as weighted Euler characterstics. Because of my own interest in abelian threefolds, I aim at covering enough of the foundations to be able to explain that properly phrased, DT invariants for abelian threefolds are meaningful and nontrivial.
Friday, March 19th Henning Lohne (UiB)
Title: Minimal free resolutions of some monomial ideals
Abstract:
We are interested in describing the minimal free resolution of a monomial ideal explicitly. We will do this for a class of ideals called Borel (fixed) ideals using the Eliahou--Kervaire minimal free resolution and by a construction of Sinefakopoulos when the ideal is generated in one degree. These resolutions are what is called cellular and are resolutions given by a combinatorial description. If time, we will also give examples of something called box-resolutions. The talk will be an introduction to the area and will not contain any new results.
Friday, April 23rd Jan Kleppe (HiO)
Deformations of determinantal schemes and modules of maximal grade
Abstract:
La M være end.gen. gradert modul over en polynomring R. M sies å ha maksimal dybde dersom det 0-te Fitting idealet I til en homogen presentasjonsmatrise Ma for M har maksimal codimensjon i R. I dette tilfellet er I = ann(Ma) og A:=R/I er en determinansiell ring (og I er generert av de maksimale minorene til Ma).

(i) Vi viser at Hom(M,M)=A og at Exti(M,M)= 0 for i=1 (hhv. for i=2) dersom dim A > 1 (hhv. > 2) der Ext er ExtA, dvs. ser på M som A-modul. For å vise dette tar jeg for meg en nyttig teknikk for forsvinning av Ext dersom M har nok dybde.

(ii) Vi viser at den lokale deformasjonsfunktoren til M og den til A er isomorfe dersom (i) holder.

(iii) Vi konkluderer med at når dim Proj A > 1 så vil tillukningen av den delmengden i Hilb(Pn) som svarer til determinansielle skjemaer hvor gradene av entriene i Ma er fiksert, danne en generisk glatt irred. komponent av Hilb(Pn) og vi gir et eksplitt uttrykk for dimensjonen av komponenten. Resultatet gjelder i alle kodimensjoner (i kodimensjon 2 s er resultatet vist av Geir Ellingsrud).

Friday, April 30th Jon Eivind Vatne (HiB)
Examples of Koszul operads and algebras
Abstract:
An operad is basically a tool for encoding various algebraic structures, the classical cases being associative, associative commutative, and Lie algebras. In recent years, many operads with properties somewhat similar to these classical examples have been constructed. In this elementary talk, we will skip the technical operadic issues and give examples of algebras of various types, including (time-permitting) Pre-Lie, Leibniz, Permutation, Zinbiel and possibly others. We will also try to understand the motivation (if any) for introducing these classes of algebras.
Friday, May 7th Jan-Magnus Økland (UiB)
The decomposition theorem for semi-small maps and the Hilbert scheme of points on a non-singular surface.
Abstract:
For semi-small maps f : X -> Y, with Y not necessarily non-singular, the push-forward of the constant sheaf on X is still a perverse sheaf on Y. Examples include the Hilbert-Chow map S[n] -> S(n), often seen as a resolution of singularities of the n-fold symmetric product of a non-singular surface S. Following an expository paper by M.A. de Cataldo and L. Migliorini, we tell the story of lifting Macdonald's formula for H*(S(n)) to Goettsche's formula for H*(S[n])
Friday, June 4th, 12:30, room 534 (Note the time and place) Stein Arild Strømme (UiB)
Title: Noen bigraderte CM-ringer og deres resolusjoner
Abstract:
Ringene det er snakk om oppstår slik: La P1=(a1,b1), ... , Pn=(an,bn) være n distinkte punkter i det affine planet A2, og kall deres union Z. (Det tilfellet jeg studerer er at Z er diagrammet til en partisjon av n). La X \subset Zn \subset A{2n} bestå av alle n-tupler i Z som består av n distinkte punkter. X har n! elementer, som utgjør en orbit under permutasjonsvirkningen på {A2}n (der faktorene permuteres). Vi skal se på koordinatringen til X som kvotient av k[x1,...,xn, y1,...yn], men ikke bare den: den interessante ringen er den assossierte bigraderte ringen. Man lager den ved å la en maksimal rang-2 torus {(s,t)|s,t i k*} virke naturlig på A2, lager en 2-dimensjonal familie ved å ta tillukning av torus-orbiten til X, og til slutt ser vi hva denne familien blir i grensepunktet når (s,t)->(0,0). Målet er å vise at denne familien er flat, slik at den spesielle fiberen også har lengde n!.
Wednesday, June 9th, 14:15, (Note the time) room 534 Stein Arild Strømme (UiB)
Title: Noen bigraderte CM-ringer og deres resolusjoner (continuation)

Last updated June 8th, 2010 by andreas.knutsen[at]math.uib.no.