Surfaces with Many Solitary Points

2008/01/28 by Labs, Erwan Brugalle Oliver
#14J17 #14J70 #14P25 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0801.4283

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 x²+y²+z²=0) whereas it can have up to four nodes (x²+y²-z²=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≥ 1.

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