Real and complex varieties: comparisons, contrasts and examples

Institute for Mathematics and its Applications
University of Minnesota
April 26, 2007

I want to begin by thanking the IMA administrators and staff for running this incredibly excellent year-long program
on Applications of Algebraic Geometry. And in the same context the organizers of the various workshops, tutorials,
and seminars also deserve to be highly commended.


Examples and properties of plane curves

Quadric surfaces, mostly ruled ones
Quadric cone
Hyperbolic paraboloid
Hyperboloid of one sheet
A cone asymptotic to a hyperboloid
The hyperboloid and a tangent plane



Tangent surfaces of space curves.

  • The tangent surface of the twisted cubic.    
    The image of a rectangular coordinate patch
    Another view, with equal length tangent line segments

  • The tangent surface of a curve is singular at all points of the curve itself. In the case of the twisted cubic, these are the only singular points.
  • Tangent surfaces of other curves.
    • No sketches are presently available.
    • Tangent lines at two distinct points can intersect non-trivially.
      Hence, these surfaces can have other components of their singular loci
      that resemble the ordinary double points of a generic projection.

Images under generic projection.
Cubic ruled surface
The Steiner surface


  • Given a smooth surface in P4 (projective 4-space), generic projection to P³ refers to the process of centrally projecting it it from a generic point of P4. In the case of a smooth surface in P5, we project from a generic line in P5. {Or we can iterate the process of projecting from a point.}
  • The image of a smooth surface, under generic projection to P³, is a surface with a purely 1-dimensional singular locus. Most of the singular points are ordinary double points (where two smooth sheets of the surface cross transversally). There are finitely many pinch points and finitely many triple points.
  • For a complex projective variety X, a corollary of the Fulton-Hansen connectedness theorem says that pinch points always occur when X is projected to Pn, where n < 2·dim(X) - 1. {The proof of this corollary involves observing that the pre-image of the diagonal is connected if we map XxX to PnxPn.}
  • There are very few examples of generic projections of smooth surfaces to P³ where triple points are not present. Our cubic ruled surface is one of them; it is the generic projection of a smooth surface of degree 3 in P4.
  • About the real locus.
  • The entire z-axis is part of the real locus of the cubic ruled surface. But only the part with -1 < z < 1 is shown. Originally, I did this because this part indicates more clearly the nature of the complex surface.
  • The other portions of the z-axis, however, indicate an interesting feature of some singular real varieties:
    In the neighborhood of a singular point, the local dimension of a real variety can be lower than what you might expect.
      Projective duality About the pictures