Equations and syzygies of (abelian) surfaces.
Equations, and more generally syzygies of projective varieties express a deep relation between the geometry of the variety and the algebra of its coordinate ring. For example, we can use the presence of an unexpected syzygy to single out interesting loci in moduli spaces of polarized varieties. On the other hand, we can also study uniform good properties of the syzygies for embeddings by very positive line bundles. In recent years, much work has been done on the case of curves and K3 surfaces, and in this talk we aim to introduce this circle of ideas in the case of abelian surfaces.