Classical algebraic geometry begins by interpreting the solutions to polynomial equations as geometric objects. The solutions can then be studied geometrically, and a correspondence between their algebra and geometry is established. Through this correspondence, geometric properties can sometimes be derived algebraically.
In this case, however, we defined a geometric object starting from an algebraic object. My question is, what are some examples where the algebraic side of algebraic geometry can be applied to understand geometric objects which are not a priori defined based off of some algebraic object? I imagine, for example, that there are geometric objects that turned out to be algebraic varieties although they were originally motivated some other way. A more interesting example might be a geometric object that isn't an algebraic variety, but where algebraic techniques from algebraic geometry can still be applied.