Cecilia Trifogli

Focal Loci of Algebraic Varieties I

The focal locus ∑x of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x ∈X and a circle with centre O passing through x which osculates X inx. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map ∈ between the Euclidean normal bundle Nx and the projective ambient space (∈ sends the normal vector O - x to its endpoint O), and in this paper we address two general problems:. 1)Characterize thedegeneratecase where the focal locus is not a hyper surface. 2)Calculate, in the case where ∑x is a hypersurface, its degree (with multiplicity).

Focal Loci of Algebraic Hypersurfaces: A General Theory

AbstractThe focal locus is traditionally defined for a differentiable submanifold of Rn. However, since it depends essentially only on the notion of orthogonality, a focal locus can be also associated to an algebraic subvariety of the space n

Thomas Wylton on Motion

P_C^n

Thomas Wylton Against Minimal Times

n, once we have chosen an orthogonal structure on this space. In this paper, we establish somebasic results in the theory of focal loci of algebraichypersurfaces in n

Robert Kilwardby on Time

P_C^n

Giles of Rome on Natural Motion in the Void

n. Our main results concern the irreducibility of the ramification divisor of the end-point map and the dimension of the singular locus of this divisor, the birationality of the focal map and the degree of the focal locus of an algebraic hypersurface.