Toric stacks I: The theory of stacky fans
Toric stacks I: The theory of stacky fans
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Toric Stacks I：堆叠扇理论
DOI：
10.1090/s0002-9947-2014-06063-7
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发表时间：
2011
影响因子：
1.3
通讯作者：
M. Satriano
中科院分区：
文献类型：
--
作者：
Anton Geraschenko;M. Satriano
本文及其后续文章的目的是介绍和发展一个环堆理论，该理论包含和扩展了文献中定义的几个环堆概念，以及经典的环堆变体。
The purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties.
In this paper, we define a toric stack as the stack 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 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 P^n and [A^1/G_m]. 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 papers by Cox and Perroni, 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.