Toric stacks I: The theory of stacky fans

By Anton Geraschenko, Matthew Satriano. 2011

80 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6be42a95613d0b1c3950697276bf9061f1b647d5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Martin Ulirsch
2. Renzo Cavalieri
3. T. Kuwagaki
4. Jonathan Wise
5. Melody Chan
6. David Zureick-Brown
7. Matthew Satriano
8. Jordan S. Ellenberg

In Discrete Mathematics and Combinatorics

1. T. Kuwagaki
2. Martin Ulirsch
3. Jonathan Wise
4. Renzo Cavalieri
5. Jordan S. Ellenberg

In Geometry and Topology

1. Martin Ulirsch
2. Renzo Cavalieri
3. T. Kuwagaki
4. Melody Chan
5. Jonathan Wise
6. David Zureick-Brown
7. Matthew Satriano
8. Jordan S. Ellenberg

Citations in the corpus

• The nonequivariant coherent-constructible correspondence for toric stacks (2015, background): “of γ∨ 0 . Then, the statement can be shown by an argument similar to the previous paragraph. 9 4 Preliminaries on toric stack For generalities on toric stacks, we refer to Geraschenko and Satoriano’s [GS15]. In this paper, we will use a special class of toric stacks which includes any toric varietie”
• A MODULI STACK OF TROPICAL CURVES (2017, background): “plit diagonalizable groups. Its topology is generated by torus-invariant open subsets and, taking Pbe the class of smooth toric morphisms, we obtain a geometric context. Toric stacks (in the sense of [GS15]) are then nothing but geometric stacks in this context. A geometric groupoid is a groupoid ob”
• Autoequivalences of derived categories via geometric invariant theory (2013, unknown): “(We use a very simple case of the stacky fan construction from [12].”
• Heights on stacks and a generalized Batyrev–Manin–Malle conjecture (2021, background): “A weighted projective space is an example of a toric stack, as in [32].”