Gary Kennedy,
Tangent Star Cones

Suppose that x is a point on X, an affine algebraic variety. The tangent star to X at x consists of all limits of secant lines L(p,q), as p and q independently approach x. Its projectivization is the fiber over x of the normal cone of X inside the self-product X times X, called the tangent star cone. With this description, one sees that the tangent star cone doesn't depend on the embedding. Furthermore, it has a natural scheme-theoretic structure, and may well be nonreduced even when X itself is reduced. Besides its naive appeal, the TS cone plays a prominent role in intersection theory. For a flat family of varieties, the TS cones will likewise form a flat family, if the Segre classes of the relative TS cone specialize to those of the fibers, and if the relative TS cone is Cohen-Macaulay. Conjecture: For a relative local complete intersection, the relative TS cone is Cohen-Macaulay.