Algebraic Varieties

Compared to manifolds, algebraic varieties are very diverse even locally. A possible entry point into the theory of algebraic varieties is the comparison with submanifolds of real or complex vector spaces. We have obtained examples of such submanifolds as zero sets of differentiable functions, whose derivative on the zero set is everywhere different from zero. If one considers polynomial functions instead of differentiable functions, one can also consider such zero sets for other fields. The “algebraic” in algebraic varieties refers to the fact that the functions considered have a polynomial character.