An affine algebraic variety is an irreducible closed subset of (with the induced topology) as discussed in the previous two posts.
A non-constant polynomial in the variables and defines a plane curve affine variety . Below is the variety for the plane elliptic curve .
The set of four points in is an affine variety. It is the intersection of an ellipse and a hyperbola .
A quadratic is an affine algebraic variety defined by a single quadratic polynomial. In these are the plane conics; circles, ellipses, parabolas, and hyperbolas in . In these are; spheres, ellipsoids, paraboloids, and hyperboloids in .
Below are examples of the two paraboloid affine algebraic varieties, a hyperbolic paraboloid (a Pringle has this surface!) and an elliptic paraboloid .
Below are examples of two hyperboloid affine algebraic varieties, a hyperboloid of one sheet and a hyperboliod of two sheets .
Plane curves and quadratics defined by a single polynomial are called hyper-surfaces. Some affine algebraic varieties are intersections of hyper-surfaces.
Below is an example of an affine algebraic variety, the intersection of two quadratics and that is a `figure of eight’ curve in .
Irreducible varieties such as the plane curve in and the surface in are irreducible hyper-surfaces. These are visualised below; the first has components that are connected in Euclidean topology, the second has five components that meet at four singular points.