M.I.T. Algebraic Geometry Seminar - Brendan Hassett
Friday March 5 , 4:00pm-5:30pm
Room 2-255
Equivariant compactifications of the additive group
(joint work with Yuri Tschinkel)
Toric varieties (normal equivariant compactifications of the multiplicative
group) have been extensively studied and are widely used in algebraic
geometry. On the other hand, equivariant compactifications of the additive
group have attracted very little attention. They are more complicated than
toric varieties in two respects. First, they are not determined by
combinatorial data and may even admit moduli. Second, even simple
varieties (like projective spaces) admit many different structures as
equivariant compactifications of the additive group. To study
these phenomena, we give a dictionary relating Artinian local rings, certain
systems of partial differential equations, and equivariant compactifications
of the additive group. We also prove some fundamental geometric
results about these compactifications. As an application, we
describe two-dimensional (and some three-dimensional) smooth compactifications
in detail; we also classify the structures on projective spaces of
small dimension.
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