For complex projective varieties, all natural transformations from constructible functions to homology (modulo torsion) are linear combinations of the MacPherson-Schwartz-Chern classes. In particular, the total Chern class is the only such natural transformation c such that for all projective spaces P, the top component of c(1P) is the fundamental class of P.

The natural mapping of the right quaternion vector space H onto the quaternion projective line (identified with the four-sphere) can be defined for complex quaternions H ⊗R C as well. We discuss its exceptional set, the fiber subspaces, and how the linear automorphism groups of two-dimensional quaternion vector spaces and modules induce groups of projective automorphisms of the image quadrics.

2 Vector Spaces and Projective Spaces 3 2.1 Vector spaces and their duals . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Projective spaces and homogeneous coordinates . . . . . . . . . . . . . . . 5 2.2.1 Visualizing projective space . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Homogeneous coordinates . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Linear subspaces . . . . ....

We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bézout Equality for equi–dimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As ...