Algebraic Variety

A generalization to $n$-D of Algebraic Curves. More technically, an algebraic variety is a reduced Scheme of Finite type over a Field $K$. An algebraic variety $V$ is defined as the Set of points in the Reals $\Bbb{R}^n$ (or the Complex Numbers $\Bbb{C}^n$) satisfying a system of Polynomial equations $f_i(x_1, \ldots, x_n)=0$ for $i=1$, 2, .... According to the Hilbert Basis Theorem, a Finite number of equations suffices.

See also Abelian Variety, Albanese Variety, Brauer-Severi Variety, Chow Variety, Picard Variety

References

Ciliberto, C.; Laura, E.; and Somese, A. J. (Eds.). Classification of Algebraic Varieties. Providence, RI: Amer. Math. Soc., 1994.