806 research outputs found
Canonical Artin Stacks over Log Smooth Schemes
We develop a theory of toric Artin stacks extending the theories of toric
Deligne-Mumford stacks developed by Borisov-Chen-Smith, Fantechi-Mann-Nironi,
and Iwanari. We also generalize the Chevalley-Shephard-Todd theorem to the case
of diagonalizable group schemes. These are both applications of our main
theorem which shows that a toroidal embedding X is canonically the good moduli
space (in the sense of Alper) of a smooth log smooth Artin stack whose stacky
structure is supported on the singular locus of X.Comment: To appear in Mathematische Zeitschrif
de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities
We prove that the Hodge-de Rham spectral sequence for smooth proper tame
Artin stacks in characteristic p (as defined by Abramovich, Olsson, and
Vistoli) which lift mod p^2 degenerates. We push the result to the coarse
spaces of such stacks, thereby obtaining a degeneracy result for schemes which
are etale locally the quotient of a smooth scheme by a finite linearly
reductive group scheme.Comment: Final version, to appear in Annales de l'Institut Fourie
Toric Stacks I: The Theory of Stacky Fans
The purpose of this paper and its sequel (Toric Stacks II) is to introduce
and develop a theory of toric stacks which encompasses and extends the notions
of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well
as classical toric varieties.
In this paper, we define a \emph{toric stack} as a quotient of a toric
variety by a subgroup of its torus (we also define a generically stacky
version). Any toric stack arises from a combinatorial gadget called a
\emph{stacky fan}. We develop a dictionary between the combinatorics of stacky
fans and the geometry of toric stacks, stressing stacky phenomena such as
canonical stacks and good moduli space morphisms.
We also show that smooth toric stacks carry a moduli interpretation extending
the usual moduli interpretations of and
. Indeed, smooth toric stacks precisely solve
moduli problems specified by (generalized) effective Cartier divisors with
given linear relations and given intersection relations. Smooth toric stacks
therefore form a natural closure to the class of moduli problems introduced for
smooth toric varieties and smooth toric DM stacks in [Cox95] and [Per08],
respectively.
We include a plethora of examples to illustrate the general theory. We hope
that this theory of toric stacks can serve as a companion to an introduction to
stacks, in much the same way that toric varieties can serve as a companion to
an introduction to schemes.Comment: 36 pages (update to match published version
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