what can the geometry of a moduli space tell us? suppose we look at the set of all subspaces in projective space. we can turn this into an geometric object (the hilbert scheme) whose points correspond to subspaces of projective space. but what does the geometry tell us? in other words, what do the connected components represent? what about the tangent space? what if it has singularities? what do these things tell us about the subspaces we're parametrizing? what if we map a curve into this space: does that give us anything meaningful? i will discuss these concepts through examples. it turns out the hilbert scheme is a very robust object, but it comes at a price: which will be encapsulated in a theorem aptly called "murphy's law for hilbert schemes".