69 research outputs found
Towards non-reductive geometric invariant theory
We study linear actions of algebraic groups on smooth projective varieties X.
A guiding goal for us is to understand the cohomology of "quotients" under such
actions, by generalizing (from reductive to non-reductive group actions)
existing methods involving Mumford's geometric invariant theory (GIT). We
concentrate on actions of unipotent groups H, and define sets of stable points
X^s and semistable points X^{ss}, often explicitly computable via the methods
of reductive GIT, which reduce to the standard definitions due to Mumford in
the case of reductive actions. We compare these with definitions in the
literature. Results include (1) a geometric criterion determining whether or
not a ring of invariants is finitely generated, (2) the existence of a
geometric quotient of X^s, and (3) the existence of a canonical "enveloping
quotient" variety of X^{ss}, denoted X//H, which (4) has a projective
completion given by a reductive GIT quotient and (5) is itself projective and
isomorphic to Proj(k[X]^H) when k[X]^H is finitely generated.Comment: 37 pages, 1 figure (parabola2.eps), in honor of Bob MacPherson's 60th
birthda
A^1-homotopy groups, excision, and solvable quotients
We study some properties of A^1-homotopy groups: geometric interpretations of
connectivity, excision results, and a re-interpretation of quotients by free
actions of connected solvable groups in terms of covering spaces in the sense
of A^1-homotopy theory. These concepts and results are well-suited to the study
of certain quotients via geometric invariant theory.
As a case study in the geometry of solvable group quotients, we investigate
A^1-homotopy groups of smooth toric varieties. We give simple combinatorial
conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy
groups of smooth (proper) toric varieties. Finally, in certain cases, we can
actually compute the "next" non-vanishing A^1-homotopy group (beyond
\pi_1^{A^1}) of a smooth toric variety. From this point of view, A^1-homotopy
theory, even with its exquisite sensitivity to algebro-geometric structure, is
almost "as tractable" (in low degrees) as ordinary homotopy for large classes
of interesting varieties.Comment: 48 pages, To appear Adv. Math, typographical and grammatical update
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