Proposition

Sketch the following affine varieties in $\mathbb{R}^2$:

1. $V(x^2 + 4y^2 + 2x - 16y + 1)$.
2. $V(x^2 - y^2)$.
3. $V(2x + y - 1, 3x - y + 2)$.

Solution

1

$x^2 + 4y^2 + 2x - 16y + 1 = 0 \implies (x + 1)^2 + 4(y - 2)^2 = 4^2$. Thus it is an ellipse centered at $(-1, 2)$ and radii $4, 2$.

We have 1 equation in a 2-dimensional space, so it is intuitive that it is a 1-manifold.

2

$\{ (t, \pm t) \mid t \in \mathbb{R} \}$.

We have 1 equation in a 2-dimensional space, so it is intuitive that it is a 1-manifold.

3

$(x, y) = (-1/5, 7/5)$ is the only point that satisfies the two polynomials.

We have 2 equations in a 2-dimensional space, so it is intuitive that it is a 0-dimensional space.