Derived categories of Gushel-Mukai surfaces and Fano fourfolds of K3 type
Yulieth Prieto-Montañez, Ian Selvaggi
TL;DR
The paper addresses how GM surfaces connected by duality can be derived- and L-equivalent without being isomorphic, and it extends these ideas to Fano fourfolds of K3 type by comparing two HPD-derived semiorthogonal decompositions. It proves that very general dual GM surfaces are D- and L-equivalent yet nonisomorphic, and it identifies a K3 surface Z arising in the K3-38 Fano fourfold with the Jacobian-based dual T, establishing nonisomorphism despite equivalence. Through HPD and Jacobian analysis, it shows that the two Kuznetsov components in the SODs for X are FM partners that are nonisomorphic, illuminating subtle interactions between D- and L-equivalence in hyperkähler-related geometry. These results provide new instances of FM-partner phenomena in GM/Fano-K3 contexts and clarify the HPD relations that govern derived categories in this setting.
Abstract
We prove that very general, dual Gushel-Mukai surfaces are not isomorphic, though derived and L-equivalent. We use this result to study two semiorthogonal decompositions for a family of Fano fourfolds of K3 type, answering a question by Bernardara-Fatighenti-Manivel-Tanturri.