Speaker: Fabrizio Catanese
Title: Focal varieties of algebraic varieties.
Affiliation: Goettingen
Date: Friday, October 22.
Place and Time: Room 101 - Love Building, 3:30-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. Given an Euclidean space E (an affine space with a non degenerate quadratic form), and a submanifold M, one defines in differential geometry the focal locus F of M as the locus of centres of curvatures of M. One can also define F as the locus of points where the infinitely near normal spaces meet ( i.e. , the ramification locus of the map from the normal bundle N_{M} to E ). In the case of a plane curve M , F is the so called evolute of M ( for the ellipse we get the astroid). The classical work dealt extensively with the case of plane curves and of surfaces in 3-space, especially of degree 2 ( work of Salmon, Fiedler, lectures by Hilbert). Recently, the case of curves and hypersurfaces was reconsidered in the work of Fantechi and Trifogli, who extended greatly classical formulae and results. In joint work with Trifogli, we are laying down the foundation of the general theory for higher codimension algebraic varieties (in this case F is also an algebraic variety !). For instance, we show that if we move M by a general projectivity, then F becomes a hypersurface whose degree is nicely calculated via the theory of Chern classes. I will also mention work in progress towards describing the degenerate cases where F is not a hypersurface ( for plane curves, that is only lines and circles ! but already in 3 - space we get many more examples).