Star points on smooth hypersurfaces

Abstract

A point P on a smooth hypersurface X of degree d in P N is called a star point if and only if the intersection of X with the embedded tangent space T P (X) is a cone with vertex P . This notion is a generalization of total inflection points on plane curves and Eckardt points on smooth cubic surfaces in P 3 . We generalize results on the configuration space of total inflection points on plane curves to star points. We give a detailed description of the configuration space for hypersurfaces with two or three star points. We investigate collinear star points and we prove that the number of star points on a smooth hypersurface is finite.

Computing α-Invariants of Singular del Pezzo Surfaces

“…The equality lct n (X) = 1 − 1/m holds ⇐⇒ the hypersurface X contains a cone of dimension m − 2 (see [2,Theorem 1.3], [2,Theorem 4.1], [13,Theorem 0.2]). Then lct n,2 X > dim(X) dim(X) + 1 by Remark 1.8, [2, Remark 1.6], [2, Theorem 4.1], [2, Theorem 5.2] and [13, Theorem 0.2], because X contains at most finitely many cones by [9,Theorem 4.2]. If X is general, then…”

On the moduli space of pairs consisting of a cubic threefold and a hyperplane

“…Eckardt points are also called star points by other authors (cf. [CC10]). If V is smooth, then the second condition is equivalent to saying that (V ∩ T p V ) ⊂ T p Y is a cone with vertex p over a (n − 2)-dimensional cubic hypersurface.…”

Singular hypersurfaces possessing infinitely many star points

“…Let Q be a general point on C P . The proof of [2,Theorem 4.2] implies that the line P, Q contains a singular point R of X. As a matter of fact, in the case of smooth hypersurfaces this holds for any component of T P (X) ∩ X using the fact that no such component is a cone with vertex T P (Λ).…”

“…A smooth point on X is called a star point if and only if the intersection of X with the embedded tangent space T P (X) is a cone with vertex P . As explained in [2], this notion is a generalisation of total inflection points on plane curves. It is also a generalisation of the classical notion of an Eckardt point on a smooth cubic surface in P 3 (see [3,8,9]).…”

“…It is also a generalisation of the classical notion of an Eckardt point on a smooth cubic surface in P 3 (see [3,8,9]). Star points on smooth hypersurfaces have been studied in [2], where it is proven that such hypersurfaces contain only finitely many star points. Star points on singular cubic surfaces in P 3 have been examined in [1,9,10].…”