Radical Generic Initial Ideals

In this paper, we survey the theory of Cartwright–Sturmfels ideals. These are ℤn-graded ideals, whose multigraded generic initial ideal is radical. Cartwright–Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Gröbner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright–Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright–Sturmfels.