Quotients of incidence geometries

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also explore geometric properties such as connectivity, firmness and transitivity conditions to determine when they are preserved under the quotienting operation. We show that the class of coset pregeometries, which contains all flag-transitive geometries, is closed under an appropriate quotienting operation.Comment: 26 pages, 5 figure

Basic and degenerate pregeometries

We study pairs $(\Gamma,G)$, where $\Gamma$ is a 'Buekenhout-Tits' pregeometry with all rank 2 truncations connected, and $G\leqslant\mathrm{Aut} \Gamma$ is transitive on the set of elements of each type. The family of such pairs is closed under forming quotients with respect to $G$-invariant type-refining partitions of the element set of $\Gamma$. We identify the 'basic' pairs (those that admit no non-degenerate quotients), and show, by studying quotients and direct decompositions, that the study of basic pregeometries reduces to examining those where the group $G$ is faithful and primitive on the set of elements of each type. We also study the special case of normal quotients, where we take quotients with respect to the orbits of a normal subgroup of $G$. There is a similar reduction for normal-basic pregeometries to those where $G$ is faithful and quasiprimitive on the set of elements of each type

Problems on Polytopes, Their Groups, and Realizations

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete Geometry, to appear

The moduli space of stable quotients

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill-Noether constructions. The moduli space of stable quotients is proven to carry a canonical 2-term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the Gromov-Witten theory of the Grassmannian in all genera. Stable quotients can also be used to study Calabi-Yau geometries. The conifold is calculated to agree with stable maps. Several questions about the behavior of stable quotients for arbitrary targets are raised.Comment: 50 page

Regular Incidence Complexes, Polytopes, and C-Groups

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder, A. Deza, and A. Ivic Weiss (eds), Springe