A wall-crossing formula for degrees of Real central projections

Christian Okonek (1) , Andrei Teleman (2)

1 Institut für Mathematik [Zürich]
2 I2M - Institut de Mathématiques de Marseille

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 ℤ-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.

Mots clés: Real projective variety; degree; central projection; relative orientation; wall-crossing; conservation of numbers

Domaines: Géométrie différentielle [math.DG]; Géométrie algébrique [math.AG]; Topologie algébrique [math.AT]

Christian Okonek, Andrei Teleman. A wall-crossing formula for degrees of Real central projections. International Journal of Mathematics, 2014, pp.1450038. ⟨10.1142/S0129167X14500384⟩. ⟨hal-01104677⟩