The picard invariant of a principal homogeneous space

The Picard Group of a Loop Space

The loop space LP1 of the Riemann sphere consisting of all C k or Sobolev W k,p maps S → P1 is an infinite dimensional complex manifold. We compute the Picard group Pic(LP1) of holomorphic line bundles on LP1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H(LP1). The group of Möbius transformations G and its loop group LG act on LP1. We prove that an element o...

The Rasmussen Invariant of a Homogeneous Knot

A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot. In particular, we recover Baader’s theorem which states that a knot is posi...

The Picard-Vessiot Theory of Homogeneous Linear Ordinary Differential Equations.

On the Geometry of Principal Homogeneous Spaces

Let k be an algebraically closed field, let π : X → B be an elliptic surface defined over k, and let XK be the generic fiber of π, which is an elliptic curve defined over the field K = k(B), the function field of B. If f : Y → B is a genus one fibration locally isomorphic to X (in the étale topology on B), then Y corresponds to a principal homogeneous space YK over XK which is everywhere locall...

On the Geometry of Principal Homogeneous Spaces

Let B be a curve defined over an algebraically closed field k and let π: X → B be an elliptic surface with base curve B. We investigate the geometry of everywhere locally trivial principal homogeneous spaces for X, i.e., elements of the Tate-Shafarevich group of the generic fiber of π. If Y is such a principal homogeneous space of order n, we find strong restrictions on the Pn−1 bundle over B i...