Forum of Mathematics Sigma最新文献

We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of $(W,c)$-polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of $(W,c)$-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.

Let $overline {mathcal {M}}_{g, m|n}$ denote Hassett’s moduli space of weighted pointed stable curves of genus g for the heavy/light weight data $$begin{align*}left(1^{(m)}, 1/n^{(n)}right),end{align*}$$

and let $mathcal {M}_{g, m|n} subset overline {mathcal {M}}_{g, m|n}$ be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which computes the generating function for $(S_mtimes S_n)$-equivariant Hodge–Deligne polynomials of these spaces in terms of the generating functions for $S_{n}$-equivariant Hodge–Deligne polynomials of $overline {mathcal {M}}_{g,n}$ and $mathcal {M}_{g,n}$.

In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topology of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures concerning tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ from the point of view of moduli spaces. The conjectures are generalized versions of the Weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields of characteristic $p>0$ which was formulated by Tamagawa. Moreover, we prove that the conjectures hold for certain points lying in the moduli space of curves of genus $0$.

We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures.

Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.

We prove that, for any countable acylindrically hyperbolic group G, there exists a generating set S of G such that the corresponding Cayley graph $Gamma (G,S)$ is hyperbolic, $|partial Gamma (G,S)|>2$, the natural action of G on $Gamma (G,S)$ is acylindrical and the natural action of G on the Gromov boundary $partial Gamma (G,S)$ is hyperfinite. This result broadens the class of groups that admit a non-elementary acylindrical action on a hyperbolic space with a hyperfinite boundary action.

In the article [CEGS20b], we introduced various moduli stacks of two-dimensional tamely potentially Barsotti–Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil–Kisin modules with descent data. We study the irreducible components of these stacks, establishing, in particular, that the components of the former are naturally indexed by certain Serre weights.