Moduli Spaces and Invariant Theory

A moduli space is a space that parametrizes geometric objects. For example, elliptic curves are classified by the so-called J-invariant, so the moduli space of elliptic curves is a line (with coordinate J). More generally, there exists a moduli space, calledMg , which parametries all projective algebraic curves of genus g (equivalently, all compact Riemann surfaces of genus g). The Jacobian of a Riemann surface is a moduli space that classifies line bundles on a fixed Riemann surface. The study of moduli spaces is an old branch of algebraic geometry with an abundance of technical tools: classical invariant theory, geometric invariant theory, period domains and variation of Hodge structures, stacks, derived categories, birational geometry, intersection theory, tropical geometry, etc. But we believe that a lot can be learned by studying examples using minimal machinery, as a motivation to learn more sophisticated tools.