L‐equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

We construct non‐trivial L‐equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L‐equivalence for curves (necessarily over non‐algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L‐equivalence and derived equivalence.

The proof of the L‐equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr(2,5) , and L‐equivalence is extended from genus one curves to elliptic surfaces using the Ogg–Shafarevich theory of twisting for elliptic surfaces.

Finally, we apply our results to K3 surfaces and investigate when the two elliptic L‐equivalent K3 surfaces we construct are isomorphic, using Neron–Severi lattices, moduli spaces of sheaves and derived equivalence. The most interesting case is that of elliptic K3 surfaces of polarization degree ten and multisection index five, where the resulting L‐equivalence is new.