Quadratic Points of Surfaces in Projective 3-Space

TL;DR

This paper investigates quadratic points on surfaces in projective 3-space, analyzing their local indices and deriving global geometric properties, including minimum counts of quadratic points on certain surfaces.

Contribution

It provides a detailed analysis of the indices of quadratic points and establishes new global bounds and relations for surfaces with specific geometric conditions.

Findings

Generic elliptic surfaces have at least 6 quadratic points.

Elliptic surfaces with semi-homogeneous cubic forms have at least 2 quadratic points.

The number of quadratic points in a hyperbolic disc is odd.

Abstract

Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is 1/3 or -1/3, while the index of the line field is 1 or -1. Moreover, for an elliptic quadratic point whose cubic form is semi-homogeneous, we can use Loewner's conjecture to show that the index is at most 1. From the above local results we can conclude some global results: A generic compact elliptic surface has at least 6 quadratic points, a compact elliptic surfaces with semi-homogeneous cubic forms has at least 2 quadratic points and the number of quadratic points in a hyperbolic disc is odd. By studying the behavior of the cubic form in a neighborhood of the parabolic curve, we also obtain a relation between the indices of the quadratic points of a generic surface with non-empty elliptic and hyperbolic regions.