Surfaces with Many Solitary Points

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 $k\ge 1$.