Abstract Let G be a finite abelian subgroup of SL( n , ℂ), and suppose there exists a toric crepant resolution ϕ : X ⟶ ℂ n / G of the quotient variety ℂ n / G . Let Exc( ϕ ) = E 1 ∪⋅ ⋅ ⋅ ∪ E s be the decomposition of the exceptional set of ϕ into irreducible components. We prove that for every i there exists an open subset U i of X such that E i ⊂ U i , and U i is isomorphic to the total space of the canonical bundle ω E i $\omega_{E_{i}}$ of E i . Furthermore, X = U 1 ∪⋅ ⋅ ⋅ ∪ U s . This contributes to the collection of results aimed at solving a classical problem, i.e., to determine which submanifolds of a complex manifold have a neighborhood isomorphic to a neighborhood of the zero section of their normal bundle.
Abstract Non-cofinite integral braces of dimension two are classified, completing the classification of two-dimensional affine crystallographic groups which arise as adjoint group of a brace. It is shown that there are six infinite sequences of such braces and five sequences of associated flat affine manifolds. Rédei and Cohn obtained four isomorphism types of groups which allow a factorization into a pair of infinite cyclic groups. Only three of them arise as adjoint groups of non-cofinite integral braces. One of the six sequences of non-cofinite integral braces is exceptional in the sense that it consists of braces whose adjoint group is not factorizable into a pair of infinite cyclic groups.
We present a polyhedral surface in Euclidean 3-space with the topology of a Klein bottle such that every vertex has zero angle defect and the star of every vertex is embedded. From the perspective of metric geometry, the polyhedron can be viewed as the image of a piecewise smooth isometric immersion of a flat Klein bottle. It is apparently the first such explicit example.
Brion's Formula realizes the Laurent polynomial of lattice points in a lattice polytope P as the sum of rational functions associated to the vertices of P. In this paper, we consider the special case where P is a generalized permutohedron. We introduce a modification of the rational functions associated to the vertices of P depending on a given matroid M. Upon summing these rational functions, we show that the resulting Laurent polynomial Q_M(P) behaves in certain ways like the lattice points of P, exhibiting natural recursive and reciprocity behaviors. Furthermore, upon evaluating Q_M(P) at 1, we recover the matroid Euler characteristic of Larson, Li, Payne, and Proudfoot, thereby providing a refined approach to studying these quantities.
We study C1-regular surfaces in & Ropf;3 that admit tilings by a finite number of rigid motion congruence classes of tiles. We construct examples with various topologies and present a framework for a systematic study, mainly concentrating on monotilings. A finite edge prototile is a tile that has only a finite number of possible interfaces with adjacent copies of itself. We describe all monotilings by such tiles with three or less edges. We consider the question of whether a monohedral polyhedron can be "smoothed" to become a finite edge type tileable surface with the same graph structure, and we give an example where this is not possible. Finally we list some open problems.
