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.
 

Outline

• Examples and properties of plane curves
   
• Quadric surfaces, mostly ruled ones
   
• Tangent surfaces and their singularities
   
• Generic projections
   
• Projective duality
   
• About the pictures
  • Please  don't  do this at home.
    • Actually I feel very strongly that it's more beneficial to actively view
      the surface pictures by rotating them with the mouse, rather than just
      looking at the view that I've chosen to present.
  • The Geometry Center and JGV
  • About Java
  • "Triangulation" vs.  . . .
     

Examples and properties of plane curves

• Standard conics: ellipse, parabola, and hyperbola

  Projective closures and the line at infinity

  A family of cubics

  
   
• The hyperbola is disconnected because its projective closure intersects the line at infinity twice.
   
• Some of the real cubics, however, have the property that their projective closures are disconnected.
   
• More generally, the projective closure of the real curve y² = f(x) has connected components
  that correspond to the intervals where f(x) > 0.
   

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

 

• The quadric cone has a unique singular point, namely its vertex.
   
• Each of the smooth quadric ruled surfaces contains two families of straight lines.
   
• The intersection of the hyperboloid and the tangent plane is a reducible plane conic
   -- accordingly, the union of two lines in the tangent plane.
   
• Every smooth quadric over C is ruled.
   
• Non-ruled smooth (real) quadrics: ellipsoid, elliptic paraboloid, hyperboloid of two sheets.
  • Sketches not currently available
  • The intersection of each surface with a tangent plane is a (singular) conic with only one real point.

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

 

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 XX to PⁿPⁿ.}