We examine in detail the stable reduction of $G$-Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $\left(0,p\right)$, where $G$ has a cyclic $p$-Sylow subgroup of order $p^n$. If $G$ is further assumed to be $p$-solvable (that is, $G$ has no nonabelian simple composition factors with order divisible by $p$), we obtain the following consequence: Suppose $f:Y\to {\mathbb{P}}^1$ is a three-point $G$-Galois cover defined over $ℂ$. Then the $n$-th higher ramification groups above $p$ for the upper numbering for the extension $K∕\mathbb{Q}$ vanish, where $K$ is the field of moduli of $f$. This extends work of Beckmann and Wewers. Additionally, we completely describe the stable model of a general three-point $\mathbb{Z}∕p^n$-cover, where $p>2$.

We explain how compactifications of Kuga families of abelian varieties over PEL-type Shimura varieties, including for example all those products of universal abelian schemes, can be constructed (up to good isogenies not affecting the relative cohomology) by a uniform method. We also calculate the relative cohomology and explain its various properties crucial for applications to the cohomology of automorphic bundles.

For an acyclic quiver $Q$, we solve the Clebsch–Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of $Q$ (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms between quivers and categorical Möbius inversion.

We study projectivizations of a special class of toric vector bundles that includes cotangent bundles whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of ${\overline{M}}_{0,n}$.

Let $n\ge 4$. In this article, we will determine the asymptotic behavior of the size of the set of integral points $\left(a_0:\cdots :a_n\right)$ on the hyperplane ${\sum }_{i=0}^n{X}_i=0$ in ${\mathbb{P}}^n$ such that $a_i$ is squareful (an integer $a$ is called squareful if the exponent of each prime divisor of $a$ is at least two) and $\left|a_i\right|\le B$ for each $i\in \left\{0,\dots ,n\right\}$, when $B$ goes to infinity. For this, we will use the classical Hardy–Littlewood method. The result obtained supports a possible generalization of the Batyrev–Manin program to Fano orbifolds.

We prove a denominator identity for nontwisted affine Lie superalgebras with zero dual Coxeter number.