519 research outputs found
The main component of the toric Hilbert scheme
Let \X be an affine toric variety under a torus \T and let T be a subtorus.
The general T-orbit closures in \X and their flat limits are parametrized by
the main component H_0 of the toric Hilbert scheme. Further, the quotient torus
\T/T acts on H_0 with a dense orbit. We describe the fan of this toric variety;
this leads us to an integral analogue of the fiber polytope of Billera and
Sturmfels. We also describe the relation of H_0 to the main component of the
inverse limit of GIT quotients of \X by T.Comment: 18 page
Cox rings and pseudoeffective cones of projectivized toric vector bundles
We study projectivizations of a special class of toric vector bundles that
includes cotangent bundles, whose associated Klyachko filtrations are
particularly simple. For these projectivized bundles, we give generators for
the cone of effective divisors and a presentation of the Cox ring as a
polynomial algebra over the Cox ring of a blowup of a projective space along a
sequence of linear subspaces. As applications, we show that the projectivized
cotangent bundles of some toric varieties are not Mori dream spaces and give
examples of projectivized toric vector bundles whose Cox rings are isomorphic
to that of M_{0,n}.Comment: 20 pages. v2: Revised and expanded, treating a larger class of toric
vector bundles and giving new examples whose Cox rings are related to that of
M_{0,n
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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