Height functions on Whitney umbrellas
  We study the singularities of the members of the family of height functions on Whitney umbrellas, which is also known as cross-caps, and show that the family of the height functions is a versal unfolding. Moreover, we study local intersections of a Whitney umbrella with a hyperplane through its singular point. 
Toshizumi Fukui, Masaru Hasegawa
math.DG