From Notices of the AMS
The Tour Ahead
The basic objects of algebraic geometry, such as subvarieties of a projective space, are defined by polynomial equations. The seemingly innocuous observation that one can vary the coefficients of these equations leads at once to unexpectedly deep questions:
- When are objects with distinct coefficients equivalent?
- What types of geometric objects appear if those coefficients move "towards infinity"?
- Can we make sense of "equivalence classes at infinity"?
The construction of compact moduli spaces and the study of their geometry amounts nowadays to a busy and central neighborhood of algebraic geometry. Any vibrant district in an old city, of course, has too many landmarks to visit, and the first job of a tour guide is to curate a selection of sites and routes — including multiple routes to the same site for the different perspectives they afford. Our tour today has three main stops: elliptic curves, Picard curves (together with "points on a line," their alter ego), and a brief panoramic glimpse of the general theory.Read more »