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 of space curves.
Images under generic projection.
don't do this at home.
the surface pictures by rotating them with the mouse,
rather than just
looking at the view that I've chosen to present.
Standard conics:
ellipse, parabola,
and hyperbola
Projective closures
and the line
at infinity
A family of cubics
that correspond to the intervals where
f(x) > 0.
Quadric cone
Hyperbolic paraboloid
Hyperboloid of one sheet
A cone asymptotic to a hyperboloid
The hyperboloid and a tangent plane
-- accordingly, the union of two lines in the tangent plane.
The image of a
rectangular coordinate patch
Another view, with
equal length tangent line segments
Hence, these surfaces can have other components of their singular loci
that resemble the ordinary double points of a generic projection.
Cubic ruled surface
The Steiner surface
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
don't do this at home.
the surface pictures by rotating them with
the mouse, rather than just
looking at the view that I've
chosen to present.
http://geom.math.uiuc.edu/
http://geom.math.uiuc.edu/java/JGV/