Interactive drawings of algebraic surfaces

Department of Mathematics and Statistics
University of Minnesota at Duluth
April 16, 2009

First of all, thanks very much for inviting me to speak here today. I've always enjoyed visiting Duluth, and I always appreciate opportunities to tell people about this material -- be they fellow mathematicians, students, or interested members of the general public.

We can say that algebraic geometry is about studying solution sets of systems of polynomial equations. These solution sets are called algebraic varieties. In the cases of plane curves or surfaces in 3-space, we can actually draw pictures of these sets, and the pictures can contribute significantly to our understanding.

The pictures are literally accurate only when the curve is the solution set of a polynomial equation with real coefficients and real unknowns. Nonetheless, a well drawn picture of the real locus of a complex variety can help us to understand the complex variety. Accordingly, we'll focus on pictures of curves in R² and surfaces in R³.

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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 a space curve with no singular points is the union of all of the tangent lines of the curve.

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

  • Tangent surfaces of other curves.    
The tangent surface of a rational curve of degree 4.    
  • The tangent surface of a curve is singular at all points of the curve itself.

  • In the case of the twisted cubic, the points of the curve itself are the only singular points of the tangent surface.

  • Tangent lines at two distinct points of a curve 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 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. I don't know if there are any smooth real surfaces in P4 or P5 which can be projected to P3 without acquiring pinch points.
  • 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