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From Notices of the AMS

Figure: A purple cubic surface with three $A_2$ singularities.

Algebraic and Analytic Compactifications of Moduli Spaces

by Patricio Gallardo and Matt Kerr
Communicated by Steven Sam

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"?
Searching for answers leads to the discovery of moduli spaces and their compactifications, parametrizing equivalence classes of said objects.

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.

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