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.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use TORIC STACKS I: THE THEORY OF STACKY FANS 1035This definition encompasses and extends the three kinds of toric stacks listed above:• Taking G to be trivial, we see that any toric variety X is a toric stack. • Smooth toric Deligne-Mumford stacks in the sense of [BCS05, FMN10, Iwa09] are smooth non-strict toric stacks which happen to be separated and Deligne-Mumford. See Remarks 2.17 and 2.18. • Toric stacks in the sense of [Laf02] are toric stacks that have a dense open point (i.e. toric stacks for which G = T 0 ). • A toric Artin stack in the sense of [Sat12] is a smooth non-strict toric stack which has finite generic stabilizer and which has a toric variety of the same dimension as a good moduli space. See Sections 4 and 6. • Toric stacks in the sense of [Tyo12] are toric stacks as well. This follows from the main theorem of [GS11b], stated below. See [GS11b, Remark 6.2]for more details.In this notation, the stack in Example 2.7 would be denoted [A 2 / ( 1 1 ) μ 2 ].Example 2.9. Again we have that X Σ = A 2 . This time β * = ( 1 0 ) : Z → Z 2 , which induces the homomorphism G 2 m → G m given by (s, t) → s. Therefore,We then have that X Σ,β = [(A 2 {(0, 0)})/ ( 1 1 ) G m ] = P 1 . Warning 2.11. Examples 2.6 and 2.10 show that non-isomorphic stacky fans (see Definition 3.2) can give rise to isomorphic toric stacks. The two presentations [(A 2 {(0, 0)})/ ( 1 1 ) G m ] and [P 1 /{e}] of the same toric stack are produced by different stacky fans. In Theorem B.3, we determine when different stacky fans give rise to the same toric stack.Example 2....

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack X , which specializes to the Baytev-Manin conjecture when X is a scheme and to Malle's conjecture when X is the classifying stack of a finite group.

We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S_n degree n extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions

Abstract. The purpose of this paper and its prequel 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.While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated. In the same spirit, the main result of this paper is that any Artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses.

We prove a conjecture of Medvedev and Scanlon [MS14] in the case of regular morphisms of semiabelian varieties. That is, if G is a semiabelian variety defined over an algebraically closed field K of characteristic 0, and ϕ : G → G is a dominant regular self-map of G which is not necessarily a group homomorphism, we prove that one of the following holds: either there exists a non-constant rational fibration preserved by ϕ, or there exists a point x ∈ G(K) whose ϕ-orbit is Zariski dense in G.