  We construct nontrivial 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.
