Star points on smooth hypersurfaces

TL;DR

This paper introduces the concept of star points on smooth hypersurfaces, generalizing inflection points and Eckardt points, and explores their configuration space, finiteness, and special arrangements.

Contribution

It generalizes known results on inflection points to star points on hypersurfaces and provides detailed descriptions of their configuration spaces and properties.

Findings

Number of star points on a smooth hypersurface is finite.

Detailed description of configuration space for hypersurfaces with two or three star points.

Analysis of collinear star points and their arrangements.

Abstract

A point P on a smooth hypersurface X of degree d in an N-dimensional projective space 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 three-dimensional projective space. 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.