## Visualizing affine algebraic varieties

An affine algebraic variety is an irreducible closed subset of ${A_k}^n$ (with the induced topology) as discussed in the previous two posts.

A non-constant polynomial $p(x,y)$ in the variables $x$ and $y$ defines a plane curve affine variety $Z(p) \subset {A_k}^2$. Below is the variety for the plane elliptic curve $p(x,y) := y^2-x^3+x$.

The set of four points $\{ (-2,-1),(-1,1),(1,-1),(1,2) \}$ in ${A_k}^2$ is an affine variety. It is the intersection of an ellipse $Z(x^2+y^2-xy-3)$ and a hyperbola $Z(3x^2-y^2-xy+2x+2y-3)$.

A quadratic is an affine algebraic variety defined by a single quadratic polynomial. In ${A_k}^2$ these are the plane conics; circles, ellipses, parabolas, and hyperbolas in $\mathbb{R}^2$. In ${A_k}^3$ these are; spheres, ellipsoids, paraboloids, and hyperboloids in $\mathbb{R}^3$.

Below are examples of the two paraboloid affine algebraic varieties, a hyperbolic paraboloid (a Pringle has this surface!) $Z(xy+z)$ and an elliptic paraboloid $Z(x^2+y^2-z)$.

Below are examples of two hyperboloid affine algebraic varieties, a hyperboloid of one sheet $Z(x^2-x+y^2+yz)$ and a hyperboliod of two sheets $Z(x^2+y^2-z^2+1)$.

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 $Z(x^2+y^2+z^2-1)$ and $Z(x^2-x+y^2)$ that is a `figure of eight’ curve in $\mathbb{R}^3$.

Irreducible varieties such as the plane curve $Z(y^2-x^3+x)$ in ${A_\mathbb{R}}^2$ and the surface $Z((x^2-y^2)^2-2x^2-2y^2-16z^2+1)$ in ${A_\mathbb{R}}^3$ 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.