Samuel Grushevsky, Stony Brook University

Abstract:  An n-dimensional cubic is the zero locus, in the (n+1)-dimensional projective space, of one homogeneous degree 3 polynomial in n+2 variables. Cubic curves are elliptic curves, cubic surfaces possess beautiful classical geometry (for example, a smooth cubic surface contains 27 lines), while smooth cubic threefolds were the first example (due to Clemens and Griffiths) of a variety that admits a surjective map from a projective space, but is not itself bimeromorphic to a projective space.
   
We discuss various approaches to compactifying the moduli(parameter) spaces of smooth cubic surfaces and of smooth cubic threefolds, and compare and contrast the resulting compactifications. Based on joint works with S. Casalaina-Martin, K. Hulek, R. Laza.