Starting from December 2022, I am the principal investigator of the project "Ulrich Sheaves on Hypersurfaces" funded by the call Young Researcher - SOE of MUR.
You can find an article about the project on the POLIFlash Magazine here (also available in english ).
Abstract of the project: The study of locally free sheaves supported on a projective variety is a powerful tool to understand its geometry. In this context, Ulrich sheaves play a crucial role. They are the sheaves without intermediate cohomology for which the global sections’ module has the maximal number of generators. Given a hypersurface X in the projective space, the existence of Ulrich sheaves supported on it provides significant informations about the determinantal representation of a power of the defining equation f of X (i.e. a power of f can be written as the determinant of a matrix of linear forms). Ulrich sheaves conjecturally exist on any smooth variety, although their existence has been proved in full generality only in the case of curves. The project will be developed around the study of the existence and admissible rank of Ulrich sheaves supported on hypersurfaces and it will be divided in two parts. In the first one, devoted to surfaces, we will start by dealing with the existence of low rank Ulrich sheaves on the general quintic surface in the projective space. After that, we will deal with higher degree surfaces where the goal is to determine the minimal possible rank of Ulrich sheaves. In the second part we will consider higher dimensional hypersurfaces and possibly singular ones.