Federico Tufo (Università di Bologna)

Abstract: Smooth complex Fano varieties are algbraic varieties with ample anticanonical bundle. Kollár, Miyaoka and Mori proved that in each dimensions there is a finite number of deformation families of such varieties, hence there is a hope to classify them all! We have a complete classification in dimension one, two and three, while from dimension four the problem is far to be solved. There are many tries to attack the classification in dimension four, and some of them consist in building databases of Fano fourfolds. In this talk we want to consider and compare two of them, the Kalashnikov one and the Bernardara—Fatighenti--Manivel--Tanturri one.  In the first part of the talk we introduce the tools used to build both the databases, while in the second part we explain how to translate the families of the first one to the families of the second one and viceversa. Central role of the dictionary is the relation between some special quiver moduli spaces and towers of grassmannian bundles.