We show that the singularities of spacelike maximal surfaces in Lorentz–Minkowski 3-space generically consist of cuspidal
edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter
3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap.
Keywords Maximal surfaces - Minkowski space - de Sitter space - Cuspidal cross cap
Mathematics Subject Classification (2000) Primary 57R45 - Secondary 53A10 - 53B20
Kentaro Saji was supported by JSPS Research Fellowships for Young Scientists. Masaaki Umehara and Kotaro Yamada were supported
by Grant-in-Aid for Scientific Research (No. 15340024(B)) and (No. 14340024(B)), respectively.
Dedicated to Yusuke Sakane on the occasion of his 60th birthday.