We discuss an analog of the Givental group action for the space of solutions of the commutativity equation. There are equivalent formulations in terms of cohomology classes on the Losev-Manin compactifications of genus 0 moduli spaces; in terms of linear algebra in the space of Laurent series; in terms of differential operators acting on Gromov-Witten potentials; and in terms of multi-component KP tau-functions. The last approach is equivalent to the Losev-Polyubin classification that was obtained via dressing transformations technique.
Losev and Manin introduced fine moduli spaces $\bar{L}_n$ of stable $n$-pointed chains of projective lines. The moduli space $\bar{L}_{n+1}$ is isomorphic to the toric variety $X(A_n)$ associated with the root system $A_n$, which is part of a general construction to associate with a root system $R$ of rank $n$ an $n$-dimensional smooth projective toric variety $X(R)$. In this paper we investigate generalisations of the Losev-Manin moduli spaces for the other families of classical root systems.
A root system $R$ of rank $n$ defines an $n$-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system $R$ and apply this result in the case of root systems of type $A$ to give a new proof of the fact that the toric variety $X(A_n)$ is the fine moduli space $\bar{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.
We study new effective curve classes on the moduli space of stable pointed rational curves given by the fixed loci of subgroups of the permutation group action. We compute their numerical classes and provide a strategy for writing them as effective linear combinations of F-curves, using Losev-Manin spaces and toric degeneration of curve classes.
The moduli space $\bar{M}_A$ of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of the moduli space $\bar{M}_A$ are generated by the strata of $\bar{M}_A$ and give all additive relations between them. We also observe that the Chow groups $A_i \bar{M}_A$ and the homology groups $H_{2i} \bar{M}_A$ are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data $A$.
This paper is a sequel to the paper by A. Losev and Yu. Manin [LoMa1], in which new moduli stacks $\bar{L}_{g,S}$ of pointed curves were introduced. They classify curves endowed with a family of smooth points divided into two groups, such that the points of the second group are allowed to coincide. The homology of these stacks form components of the extended modular operad whose combinatorial models are further studied in [LoMa2]. In this paper the basic geometric properties of $\bar{L}_{g,S}$ are established using the notion of weighted stable pointed curves introduced recently by B. Hassett.…
Let X be a homogeneous space, X = G/H, where G is a connected linear algebraic group over a number field k, and H is a k-subgroup of G (not necessarily connected). Let S be a finite set of places of k. We compute the Brauer-Manin obstruction to weak approximation for X in S in terms of Galois cohomology.
The spaces of point configurations on the projective line up to the action of $\mathrm{SL}(2,\mathbb K)$ and its maximal torus are canonically compactified by the Grothdieck-Knudsen and Losev-Manin moduli spaces $\overline M_{0,n}$ and $\overline L_n$ respectively. We examine the configuration space up to the action of the maximal unipotent group $\mathbb G_a\subseteq \mathrm{SL}(2,\mathbb K)$ and define an analogous compactification. For this we first assign a canonical quotient to the action of a unipotent group on a projective variety. Moreover, we show that similar to $\overline M_{0,n}$ a…
We obtain a simple, recursive presentation of the tautological (κ, ψ, and λ) classes on the moduli space of curves in genus zero and one in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, even, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann…
Cohomological field theories (CohFTs) were defined in the mid 1990s by Kontsevich and Manin to capture the formal properties of the virtual fundamental class in Gromov-Witten theory. A beautiful classification result for semisimple CohFTs (via the action of the Givental group) was proven by Teleman in 2012. The Givental-Teleman classification can be used to explicitly calculate the full CohFT in many interesting cases not approachable by earlier methods. My goal here is to present an introduction to these ideas together with a survey of the calculations of the CohFTs obtained from Witten's cla…
We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context…
We compute the (stable) étale cohomology of $\mathrm{Hom}_{n}(C, \mathcal{P}(\vecλ))$, the moduli stack of degree $n$ morphisms from a smooth projective curve $C$ to the weighted projective stack $\mathcal{P}(\vecλ)$, the latter being a stacky quotient defined by $\mathcal{P}(\vecλ) := \left[\mathbb{A}^N-\{0\}/\mathbb{G}_m\right]$, where $\mathbb{G}_m$ acts by weights $\vecλ = (λ_0, \cdots, λ_N) \in \mathbb{Z}^N_{+}$. Our key ingredient is formulating and proving the étale cohomological descent over the category $ΔS$, the symmetric (semi)simplicial category. An immediate arithmetic consequence…
We present a conjectural formula describing the cokernel of the Albanese map of zero-cycles of smooth projective varieties $X$ over $p$-adic fields in terms of the Néron-Severi group and provide a proof under additional assumptions on an integral model of $X$. The proof depends on a non-degeneracy result of Brauer-Manin pairing due to Saito-Sato and on Gabber-de Jong's comparison result of cohomological- and Azumaya-Brauer groups. We will also mention the local-global problem of the Albanese-cokernel; the abelian group on the "local side" turns out to be a finite group.
