We are concerned with the stabilizer poset of linear actions of finite groups. This is originally motivated through our attempt to describe the explicit geometry of the universal families over moduli spaces of Riemann surfaces. Here these universal families are locally approximated by linear quotient families associated with the linear actions of the automorphism groups of Riemann surfaces on the vector spaces of holomorphic quadratic differentials. To describe such families, the stabilizers for these linear actions play an important role. For instance, in these families, the fibers over stabilizer-constant loci are identical (the quotient fiber theorem). We in fact study the stabilizer posets, because they correspond to the posets of stabilizer-constant loci under the geometric Galois correspondence. We provide an algorithm to determine these stabilizer posets—in fact it works for the stabilizer posets for any linear action of any finite group. This algorithm is based on linear algebra combined with maximal conjugacy classes of stabilizers and is quite powerful in practical computation.

We introduce a new kind of welding of compact bordered Riemann surfaces, called a self-welding. We develop its fundamental theory, and apply the results to investigate, in the frame of Teichmüller theory, the set (R₀) of marked closed Riemann surfaces of positive genus into which a given marked finite open Riemann surface R₀ of the same genus can be conformally embedded. We characterize its boundary (R₀) in terms of self-welding closings of R₀.

This paper studies the Ulrich property of homogeneous vector bundles on rational homogeneous varieties. We provide a criterion for an initialized irreducible homogeneous vector bundle on a rational homogeneous variety with any Picard number to be Ulrich with respect to any polarizations. This criterion extends Fonarev's result for rational homogeneous varieties with Picard number one. As an application, we show that rational homogeneous varieties with Picard number at least two of certain exceptional algebraic groups do not admit such homogeneous Ulrich bundles with respect to the minimal ample class.

In this paper, we give the maximum of the numbers n such that we can take n simple closed geodesics without singularities that are disjoint to each other for translation surfaces in the hyperelliptic components ^hyp(2g−2) and ^hyp(g−1, g−1). The maximum is different from the maximum number of pairwise disjoint simple closed geodesics on a hyperbolic surface. We also give a classification theorem for translation surfaces in ^hyp(4) with respect to their Euclidean structures.

In this paper, we prove that hypersurface M_rⁿ with proper mean curvature vector field (i.e. Δ is proportional to ) and at most two distinct principal curvatures in a non-flat pseudo-Riemannian space form N_sⁿ⁺¹(c) is minimal or locally isoparametric, and compute the mean curvature for the isoparametric ones. As an application, we give full classification results of such non-minimal Lorentzian hypersurfaces of non-flat Lorentz space forms.