# Math 401, Fall 2025: Thesis notes, S6, Algebraic Geometry

## Affine algebraic variety

The affine algebraic variety is the common zero set of a collection $\{F_i\}_{i\in I}$ of complex polynomials on complex n-space $\mathbb{C}^n$. We write

$$
V=\mathbb{V}(\{F_i\}_{i\in I})=\{z\in \mathbb{C}^n\vert F_i(z)=0\text{ for all } i\in I\}
$$

Note that $I$ may not be countable or finite.