This talk is about complex analytic geometry, the field of mathematics concerned with complex manifolds and more generally with complex analytic spaces. The "curves" of the title are compact Riemann surfaces and the "surfaces" in the title are compact complex manifolds of dimension 2 over the complex numbers (and hence dimension 4 over the real numbers). The talk will explore the problem of constructing two dimensional complex manifolds by deforming known complex analytic spaces. It will focus on a single example. The talk should be quasi-accessible to anyone who has courses in Riemann surfaces and algebraic topology.