Towards a classification of rigid product quotient varieties of Kodaira dimension 0

In this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

Exceptional collections on certain Hassett spaces

We construct an $S_2\times S_n$ invariant full exceptional collection on Hassett spaces of weighted stable rational curves with $n+2$ markings and weights $(\frac{1}{2}+\eta, \frac{1}{2}+\eta,\epsilon,\ldots,\epsilon)$, for $0<\epsilon, \eta\ll1$ and can be identified with symmetric GIT quotients of $(\mathbb{P}^1)^n$ by the diagonal action of $\mathbb{G}_m$ when $n$ is odd, and their Kirwan desingularization when $n$ is even. The existence of such an exceptional collection is one of the needed ingredients in order to prove the existence of a full $S_n$-invariant exceptional collection on $\overline{\mathcal{M}}_{0,n}$. To prove exceptionality we use the method of windows in derived categories. To prove fullness we use previous work on the existence of invariant full exceptional collections on Losev-Manin spaces. Comment: At the request of the referee, the paper arXiv:1708.06340 has been split into two parts. This is the second of those papers (submitted). 36 pages

Measure Theoretic Entropy of Discrete Geodesic Flow on Nagao Lattice Quotient

The discrete geodesic flow on Nagao lattice quotient of the space of bi-infinite geodesics in regular trees can be viewed as the right diagonal action on the double quotient of PGL2Fq((t−1)) by PGL2Fq[t] and PGL2(Fq[[t−1]]). We investigate the measure-theoretic entropy of the discrete geodesic flow with respect to invariant probability measures.

Dense free subgroups of automorphism groups of homogeneous partially ordered sets

A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. If A is such a poset, then the group {\operatorname{Aut}(A)} has a natural topology in which {\operatorname{Aut}(A)} is a Polish topological group. We consider the problem of whether {\operatorname{Aut}(A)} contains a dense free subgroup of two generators. We show that if A is ultrahomogeneous, then {\operatorname{Aut}(A)} contains such a subgroup. Moreover, we characterize those countable ultrahomogeneous posets A such that for each natural number m the set of all cyclically dense elements {\overline{g}\in\operatorname{Aut}(A)^{m}} for the diagonal action is comeager in {\operatorname{Aut}(A)^{m}} . In our considerations we strongly use the result of Schmerl which says that there are essentially four types of countably infinite ultrahomogeneous posets.

Modular Interpretation of a Non-Reductive Chow Quotient

The space of n distinct points and adisjoint parametrized hyperplane in projective d-space up to projectivity – equivalently, configurations of n distinct points in affine d-space up to translation and homothety – has a beautiful compactification introduced by Chen, Gibney and Krashen. This variety, constructed inductively using the apparatus of Fulton–MacPherson configuration spaces, is a parameter space of certain pointed rational varieties whose dual intersection complex is a rooted tree. This generalizes $\overline M _{0,n}$ and shares many properties with it. In this paper, we prove that the normalization of the Chow quotient of (ℙd)n by the diagonal action of the subgroup of projectivities fixing a hyperplane, pointwise, is isomorphic to this Chen–Gibney–Krashen space Td, n. This is a non-reductive analogue of Kapranov's famous quotient construction of $\overline M _{0,n}$, and indeed as a special case we show that $\overline M _{0,n}$ is the Chow quotient of (ℙ1)n−1 by an action of 𝔾m ⋊ 𝔾a.

Morava K(s)*-rings of the extensions of Cp by the products of good groups under diagonal action

This note provides a theorem on good groups in the sense of Hopkins–Kuhn–Ravenel [J. Amer. Math. Soc. 13 (2000), no. 3, 553–594] and some relevant examples.

VEECH GROUPS FOR HOLONOMY-FREE TORUS COVERS

For fixed coprime k, l ∈ ℕ and each pair (w, z) ∈ ℂ2we define an infinite cyclic cover Σk,l(w, z) → 𝕋, called a k-l-surface or k-l-cover. We show that [Formula: see text] classifies k-l-covers up to isomorphism away from a rather small set. The diagonal action of SL2(ℤ) on ℂ2descends to [Formula: see text], reflecting the SL2(ℤ)-action on the family of k-l-surfaces equipped with a translation structure. The moduli space of holonomy free k-l-surfaces is a compact SL2(ℤ) invariant subspace [Formula: see text] containing all k-l-surfaces with a lattice stabilizer with respect to the SL2(ℤ) action. We calculate the stabilizer, the Veech group, explicitly and represent k-l-covers branched over two points by a generalized class of staircase surfaces. Finally we study SL2(ℤ)-equivariant translation maps from the Hurwitz space of k-(d - k)-covers to Hurwitz spaces of ℤ/d-covers branched over two points.