Surfaces with Many Solitary Points

TL;DR

This paper investigates the maximum number of solitary points on real algebraic surfaces in projective 3-space, showing it is always less than the maximum number of nodes, and constructs surfaces with many such points.

Contribution

It establishes bounds on solitary points versus nodes on surfaces and adapts existing constructions to produce surfaces with numerous solitary and other singular points.

Findings

Maximum solitary points are fewer than maximum nodes on degree ≥ 3 surfaces.

Constructed surfaces with many solitary points using refined Brusotti's theorem.

Extended constructions to surfaces with singular points of type A_{2k-1}^.

Abstract

It is classically known that a real cubic surface in the real projective 3-space cannot have more than one solitary point (locally given by x^2+y^2+z^2=0) whereas it can have up to four nodes (x^2+y^2-z^2=0). We show that on any surface of degree at least 3 in the real projective 3-space, the maximum possible number of solitary points is strictly smaller than the maximum possible number of nodes. Conversely, we adapt a construction of Chmutov to obtain surfaces with many solitary points by using a refined version of Brusotti's theorem. Finally, we adapt this construction to get real algebraic surfaces with many singular points of type $A_{2k-1}^\smbullet$ for all $k\ge 1$.