Let G be a finite abelian subgroup of SL(n, & Copf;), and suppose there exists a toric crepant resolution phi : X -> & Copf;n/G of the quotient variety & Copf;n/G. Let Exc(phi) = E1 boolean OR & sdot; & sdot; & sdot; boolean OR Es be the decomposition of the exceptional set of phi into irreducible components. We prove that for every i there exists an open subset Ui of X such that Ei subset of Ui, and Ui is isomorphic to the total space of the canonical bundle omega Ei $\omega_{E_{i}}$ of Ei. Furthermore, X = U1 boolean OR & sdot; & sdot; & sdot; boolean OR Us. 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 explicit such 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 study a modification of the rational functions associated to the vertices of P depending on a given matroid M. Upon summing these rational functions, we describe how the resulting Laurent polynomial QM(P) behaves in certain ways like the lattice points of P, exhibiting natural recursive and reciprocity behaviors. Furthermore, upon evaluating QM(P) at 1, we recover the matroid Euler characteristics of Larson, Li, Payne, and Proudfoot, so the combinatorial approach in this paper gives new insight into 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.
