Cartwright–Sturmfels Hilbert schemes

Let (Formula presented.) be the Cox ring of a product of (Formula presented.) projective spaces. In this paper, we study the Cartwright–Sturmfels Hilbert schemes of (Formula presented.), which are multigraded Hilbert schemes that parameterize only radical ideals. Our main result shows that these Hilbert schemes are always smooth and irreducible if the Picard rank (Formula presented.) is at most 2. This result can be seen as a multigraded analogue of the famous theorems of Fogarty and Maclagan–Smith, where the Picard rank replaces the dimension of the ambient space.