Logarithmic stable toric varieties and their moduli

Abstract

The Chow quotient of a toric variety by a subtorus, as defined by Kapranov-Sturmfels-Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a fixed projective toric variety, as constructed by Alexeev and Brion. We show that, after endowing both spaces with the structure of a logarithmic stack, the spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties it satisfies.

Skeletons of stable maps I: rational curves in toric varieties

“…A very closely related result appears in Chen and Satriano's description of this space in the special case where the general curve is the closure of a one‐parameter subgroup in (see [, Theorem 1.1]). Ascher and Molcho have recently generalized these results to higher rank subtori [, Theorem 1.3].…”

Tropicalization of the moduli space of stable maps

“…There are related developments by Cavalieri, Markwig, Ranganathan, Ascher, Molcho, Chen, Satriano and A. Gross [20,21,4,24,35,61].…”

Untitled

“…(2) and (3) are equivalent to saying that every cone σ that maps onto κ with relative dimension 1 can have either one or two faces isomorphic to κ; and if σ has two such faces, then N ∩ σ ∼ = (Q ∩ κ) × ‫ގ‬ ‫ގ‬ 2 for some homomorphism e q : Q ∩ κ → ‫,ގ‬ while if σ has precisely one such face, then σ ∩ N ∼ = (Q ∩ κ) × ‫.ގ‬ These statements, and the construction of the homomorphism e q are precisely the content of Lemmas 3.11 and 3.12 of [Ascher and Molcho 2015], or equivalently Lemmas 8 and 9 in [Gillam and Molcho 2013] from which the former lemmas are derived. The description given above holds étale locally on X , for the étale sheaf M X .…”

“…over a geometric point S = Spec k, with π being log smooth and flat and having reduced fibers, and where the logarithmic structure on V is a Zariski log structure. To our knowledge, these assumptions hold in all previous work where minimal logarithmic structures have been studied, e.g., [Gross and Siebert 2013;Abramovich and Chen 2014;Chen 2014;Olsson 2008;Ascher and Molcho 2015]. In fact, in these papers the authors always begin with flat, proper families of schemes with reduced fibers, and construct a minimal logarithmic structure over each geometric point by writing down an explicit formula.…”