Smooth rationally connected threefolds contain all smooth curves

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case.Comment: Version 1 was called "Any smooth toric threefold contains all curves". This version is completely rewritten and proves a much stronger result, following suggestions of Janos Kolla

Application of the Exact Muffin-Tin Orbitals Theory: the Spherical Cell Approximation

We present a self-consistent electronic structure calculation method based on the {\it Exact Muffin-Tin Orbitals} (EMTO) Theory developed by O. K. Andersen, O. Jepsen and G. Krier (in {\it Lectures on Methods of Electronic Structure Calculations}, Ed. by V. Kumar, O.K. Andersen, A. Mookerjee, Word Scientific, 1994 pp. 63-124) and O. K. Andersen, C. Arcangeli, R. W. Tank, T. Saha-Dasgupta, G. Krier, O. Jepsen, and I. Dasgupta, (in {\it Mat. Res. Soc. Symp. Proc.} {\bf 491}, 1998 pp. 3-34). The EMTO Theory can be considered as an {\it improved screened} KKR (Korringa-Kohn-Rostoker) method which is able to treat large overlapping potential spheres. Within the present implementation of the EMTO Theory the one electron equations are solved exactly using the Green's function formalism, and the Poisson's equation is solved within the {\it Spherical Cell Approximation} (SCA). To demonstrate the accuracy of the SCA-EMTO method test calculations have been carried out.Comment: 20 pages, 10 figure

Approximating curves on real rational surfaces

We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex self-intersection number

Numerical modelling of electric conductance of a thin sheet

In this paper the numeric modelling of total resistance of a thin sheet, with local conductivity in randomly distributed grains higher then is that of the basic matrix, is presented. The 2D model is formed by a structure of longitudinal and transversal conductors interconnected in nodes of a square net. In all nodes, using iteration procedure, the potential is determined from which the conductance of sheet is computed between two touching electrodes. The described model can be used to imitate the behaviour of heterogeneous thin conducting sheets prepared by different techniques. The model was verified in some cases where the net resistance is well known from the theory

KSB stability is automatic in codimension 3

KSB stability holds at codimension 1 points trivially, and it is quite well understood at codimension 2 points, since we have a complete classification of 2-dimensional slc singularities. We show that it is automatic in codimension 3.Comment: arXiv admin note: substantial text overlap with arXiv:1807.07417, arXiv:1803.0332

On Sasaki-Einstein manifolds in dimension five

We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.Comment: 18 pages, Exposition was expanded and a reference adde