A Numerical Transcendental Method in Algebraic Geometry: Computation of Picard Groups and Related Invariants

A Transcendental Method in Algebraic Geometry

It is well known that the basic objects of algebraic geometry, the smooth projective varieties, depend continuously on parameters as well as having the usual discrete invariants such as homotopy and homology groups. What I shall attempt here is to outline a procedure for measuring this continuous variation of structure. This method uses the periods of suitably defined rational differential form...

Picard Groups in Poisson Geometry

We study isomorphism classes of symplectic dual pairs P ← S → P , where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P , these Morita self-equivalences of P form a group Pic(P ) under a natural “tensor product” operation. Variants of this construction are also studied, fo...

Numerical Algebraic Geometry and Algebraic Kinematics

In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometr...

Numerical Real Algebraic Geometry

Sottile’s lectures from the Oberwolfach Seminar “New trends in algorithms for real algebraic geometry”, November 23–28, 2009.

Numerical Algebraic Geometry