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
- 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.
Images under generic projection.
Given a smooth surface in P^{4} (projective 4-space),
generic projection to P³ refers to the process of
centrally projecting it it from a generic point of P^{4}.
In the case of a smooth surface in P^{5}, we project from
a generic line in P^{5}. {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 P^{n}, 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
P^{n}P^{n}.}
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 P^{4}.
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
- Please
don't do this at home.
- The Geometry Center and JGV
- About Java
- When you view one of these figures, your computer downloads the Java
applet (a small application) from the U of MN math department
web server, along with the surface coordinates and related data from my
web page. When you rotate, translate, or scale the figure by dragging it
with your computer's mouse, the computations are done on your
computer in real time. Thus, you don't have to wait for the server
to re-draw the figure.
- "Triangulation" vs. . . .
- Although the term "triangulation" is commonly used by computer
graphics people, we're not generally referring to triangles,
but rather to other polygonal figures that we use to approximate
surfaces. Quadrilaterals are very common; for example that is
what you'd have with a parametric sketch
created in Matlab, Maple, or Mathematica. And we're often not looking
at actual planar polygons. But this turns out not to matter!!
Indeed, the edges of our polygon approximate curve segments on the
surface. These are projected onto the plane of the computer screen,
finally giving an actual polygon. This is finally filled in with whatever
color is given in my webpage data.
- If we use a relatively small grid to plot our surface, we may get
a picture that's not as smooth as what can be done by tools such as
surf. But, as mentioned above,
we are able to take advantage of the cross-platform capabilities of the
Java language, thereby obtaining figures that can be rotated much more
readily.