The purpose of this work is to apply the degeneration theory developed in [GL] to study the moduli space of stable vector bundles of arbitrary rank on any smooth algebraic surface (over C). We will show that most of the recent progress in understanding moduli of rank two vector bundles can be carried over to high rank cases. After introducing the notion of stable vector bundles, the first autho...

This paper is a continuation of our effort in understanding the geometry of the moduli space of stable vector bundles. For any polarized smooth projective surface (X,H) and for any choice of (I, d) ∈ Pic(X) × H(X,Z), there is a coarse moduli space M(I, d) of rank two μ-stable (with respect to H) locally free sheaves E of ∧E ∼= I and c2(E) = d. This moduli space has been studied extensively rece...

We prove that the lattice of linear partitions in a projective geometry of rank n has Helly number b3n/2c.

As the accuracy of biometrics improves, it is getting increasingly hard to push the limits using a single modality. In this paper, a unified approach that fuses threedimensional facial and ear data is presented. An annotated deformable model is fitted to the data and a geometry image is extracted. Wavelet coefficients are computed from the geometry image and used as a biometric signature. The m...

We investigate questions in quantum information geometry which concern the existence and non-existence of dual and dually flat structures on stratified sets of density operators on finite-dimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove moreover ...

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank 1 spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or qua...

This paper introduces a new distance and mean on the set of positive semi-definite matrices of fixed-rank. The proposed distance is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is ...

The Handbook of Incidence Geometry (Handbook of Incidence Geometry, Buildings and Foundations, North-Holland, Amsterdam, 1995) appeared in 1995. In Chapter 12, On some rank two geometries, an almost complete overview was given on the status of the theory on partial and semipartial geometries. Now, 5 years later, it is maybe a good time to give an update of this status. Indeed a lot of things ha...

With this chapter we provide a compact yet complete survey of two most remarkable “representation theorems”: every arguesian projective geometry is represented by an essentially unique vector space, and every arguesian Hilbert geometry is represented by an essentially unique generalized Hilbert space. C. Piron’s original representation theorem for propositional systems is then a corollary: it s...