A wall crossing formula for degrees of real central projections
arXiv:1206.4271
Abstract
The main result is a wall crossing formula for central projections defined on submanifolds of a real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a -valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the real subspace problem.
29 pages. First revised version: The proof of the "wall-crossing formula" is now more conceptional. We prove new general properties of the set of values of the degree map on the set of central projections. Second revised version: minor corrections. To appear in International Journal of Mathematics