Volume 20, Issue 4

The moduli space Δ g , w of tropical w -weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett’s spaces of w -weighted stable curves. If at least two of the weights are 1, we prove that Δ 0, w is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space Δ 0, w is disconnected and for which the fundamental group of Δ 0, w has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri–Hampe–Markwig–Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces Δ 1, w .

We investigate slant surfaces in the almost Hermitian manifold 𝕊 3 × ℝ, considering the position of the Reeb vector field ξ of the Sasakian structure on 𝕊 3 with respect to the surfaces. We examine two cases: ξ normal or tangent to the surfaces. In the first case, we prove that every surface is totally real. In the second case, we characterize and locally describe complex surfaces. Finally, we completely classify non-complex slant surfaces, giving explicit examples.

We construct a new family of lattice packings for superballs in three dimensions (unit balls for the l3p$\begin{array}{} \displaystyle l^p_3 \end{array}$ norm) with p ∈ (1, 1.58]. We conjecture that the family also exists for p ∈ (1.58, log 2 3 = 1.5849625…]. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has 14 neighbors.

We continue our computation, using a combinatorial method based on Gronthendieck’s dessins d’enfant , of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2 k , and local degrees over the branching points of the form [2, …, 2], [2 h + 1, 3, 2, …, 2], π =[di]i=1ℓ .$\begin{array}{} \displaystyle \pi=[d_i]_{i=1}^\ell. \end{array}$ We compute the corresponding (weak) Hurwitz numbers for several values of g and h , getting explicit arithmetic formulae in terms of the d i ’s.

By [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X . In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane 𝓗 ⊂ X , then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL 2 (ℝ). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac–Moody symmetric space G/K for an algebraically simply connected two-spherical split Kac–Moody group G , as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of α -semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

Let G be a finitely generated Fuchsian group of the first kind and let ( g : m 1 , m 2 , …, m n ) be its shortened signature. Beardon showed that almost every Dirichlet region for G has 12 g + 4 n − 6 sides. Points in ℍ corresponding to Dirichlet regions for G with fewer sides are called exceptional for G . We generalize previously established methods to show that, for any such G , its set of exceptional points is uncountable.

We present an algorithm for computing the Berkovich skeleton of a superelliptic curve y n = f ( x ) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when n is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus g .

It is proven by elementary methods that in dimension 2, every locally injective continuous map, sending the curves of a C k -spray to curves of another C k -spray as oriented point sets, is a C k -diffeomorphism. This extends the result [1] for dimension three and higher from 1965.

Let M be a real hypersurface of a complex space form M n ( c ) with c ≠ 0 and n ≥ 3. We show that the Ricci tensor S of M satisfies S ( X , Y ) = ag ( X , Y ) for all vector fields X and Y on the holomorphic distribution, a being a constant, if and only if M is a pseudo-Einstein real hypersurface. By doing this we can give the definition of pseudo-Einstein real hypersurface under weaker conditions.

We prove the existence of a linearization for singular principal G -bundles not depending on the base curve. This allow us to construct the relative compact moduli space of δ -(semi)stable singular principal G -bundles over families of reduced projective and connected nodal curves, and to reduce the construction of the universal moduli space over 𝓜 g to the construction of the universal moduli space of swamps.

We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to PSL(2, ℝ). Using this result, we describe a framework for the full classification based on the action of the group on the point set.