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 3space, 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.
Outline
To view this material later, visit:
http://www.math.umn.edu/~roberts or http://www.math.umn.edu/~roberts/Duluth_colloq 
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.




Images under generic projection.


The Cayley Surface 