For Gorenstein quotient spaces C d =G, a direct generalization of the classical McKay correspondence in dimensions d 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstei...

Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective and hyperbolic spaces, including explicit one parameter families of embeddings of quotients of certain product manifolds. In addition, new examples of minimal...

For Gorenstein quotient spaces C/G, a direct generalization of the classical McKay correspondence in dimensions d ≥ 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein...

We prove that the finite planar spaces in which, for any two disjoint planes Fl and IT, every line intersecting U intersects IT, are essentially obtained from projective spaces PG(3, k) by deleting either a line or an affino-projective (but not affine) plane of order k, or can be constructed from the Fischer spaces F1 8 and F3 6 , except perhaps if there is only one family of pairwise disjoint ...

These notes are an outline of a course on arcs given at the Finite Geometry Summer School, University of Sussex, June 26-30, 2017. Basic objects and definitions Let K denote an arbitrary field. Let Fq denote the finite field with q elements, where q is the power of a prime p. Let Vk(K) denote the k-dimensional vector space over K. Let PGk−1(K) denote the (k − 1)-dimensional projective space ove...

1 Abbreviations and Notations 3

The fundamental theorem of projective geometry is generalized for projective spaces over rings. Let RM and SN be modules. Provided some weak conditions are satisfied, a morphism g : PðMÞnE ! PðNÞ between the associated projective spaces can be induced by a semilinear map f : M ! N. These conditions are satisfied for instance if S is a left Ore domain and if the image of g contains three indepen...

85 admirably clear. While the book is devoted to a single problem, the techniques employed would seem to have wider utility, and so should appeal to those interested in efficiency of algorithms in graph theory, and further (perhaps) to those wishing to apply such ideas to the real world. I enjoyed reading this book. It is written in a way which should appeal to advanced undergraduate and postgr...