This is a request for a list of examples of problems (or other mathematical situations) that are not initially of algebro-geometric nature, but can be solved or understood by using algebraic geometry.

Here are some applications that are *not* of the kind sought:

Diophantine equations or other problems whose basic data are specified in algebraic terms, or have an immediate translation into such terms.

GAGA or reduction to finite characteristic arguments, but applied to problems that are clearly already within (or very near) the algebro-geometric sphere, involving varieties or moduli spaces, or cohomology of such spaces.