Surfaces with many solitary points

It is classically known that a real cubic surface in cannot have more than one solitary point (or -singularity, locally given by x ² + y ² + z ² = 0) whereas it can have up to four nodes (or -singularity, locally given by x ² + y ² − z ² = 0). We show that on any surface of degree d ≥ 3 in 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. Combining lower and upper bounds, we deduce: , where denotes the maximum possible number of solitary points on a real surface of degree d in . Finally, we adapt this construction to get real algebraic surfaces in with many singular points of type for all k ≥ 1.