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BEGIN:VEVENT
UID:news348@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20181204T083213
DTSTART;TZID=Europe/Zurich:20181204T103000
SUMMARY:Seminar Algebra and Geometry: Bruno Laurent (Grenoble)
DESCRIPTION:The varieties which are homogeneous under the action of an alge
braic group are very symmetric objects. More generally\, we get a much wid
er class of objects\, having a very rich geometry\, by allowing the variet
ies to have not a unique orbit\, but a dense orbit. Such varieties are sai
d to be almost homogeneous\; this includes the case of toric varities\, wh
en the group is an algebraic torus.In this talk\, I will explain how to cl
assify the pairs (X\,G) where X is a curve or a surface and G is a smo
oth and connected algebraic group acting on X with a dense orbit.For cur
ves\, I will mainly focus on the regular ones\, defined over an arbitrary
field. Over an algebraically closed field\, the "natural" notion of non-si
ngularity is "smoothness". However\, over an arbitrary field\, the weaker
notion of "regularity" is more suitable. I will recall the difference betw
een those two notions and show that there exist regular homogeneous curves
which are not smooth.For surfaces\, I will restrict to the smooth ones\,
defined over an algebraically closed field. The situation is more complica
ted than for curves. Moreover\, new phenomena and several difficulties app
ear in positive characteristic\, and I will highlight them.
X-ALT-DESC:The varieties which are homogeneous under the action of an algeb
raic group are very symmetric objects. More generally\, we get a much wide
r class of objects\, having a very rich geometry\, by allowing the varieti
es to have not a unique orbit\, but a dense orbit. Such varieties are said
to be almost homogeneous\; this includes the case of toric varities\, whe
n the group is an algebraic torus.
In this talk\, I will explai
n how to classify the pairs (X\,G) where \;X \;is a curve or a sur
face and \;G \;is a smooth and connected algebraic group acting on
\;X \;with a dense orbit.
For curves\, I will mainly f
ocus on the regular ones\, defined over an arbitrary field. Over an algebr
aically closed field\, the "\;natural"\; notion of non-singularity
is "\;smoothness"\;. However\, over an arbitrary field\, the weak
er notion of "\;regularity"\; is more suitable. I will recall the
difference between those two notions and show that there exist regular hom
ogeneous curves which are not smooth.
For surfaces\, I will res
trict to the smooth ones\, defined over an algebraically closed field. The
situation is more complicated than for curves. Moreover\, new phenomena a
nd several difficulties appear in positive characteristic\, and I will hig
hlight them.
DTEND;TZID=Europe/Zurich:20181204T120000
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