Talk presented at the special session
"Moduli spaces in Algebraic Geometry and Applications"
of the 2021 Mathematical Congress of the Americas, 15 July 2021.

Abstract: I will report on a possible characterization of Hassett divisors on the moduli space of cubic fourfolds by the property of containing special surfaces. We will discuss a construction of such special surfaces for infinitely many divisors and the relation with the work of Russo and Staglianò on the rationality of such cubics in low discriminant. Joint work with Arend Bayer (University of Edinburgh), Aaron Bertram (University of Utah), and Alex Perry (University of Michigan).