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

## TL;DR

This paper constructs new examples of L-equivalence between genus one curves, elliptic surfaces, and K3 surfaces, providing evidence for conjectures linking L-equivalence and derived equivalence.

## Contribution

It introduces the first known L-equivalence examples for curves over non-algebraically closed fields and extends L-equivalence to elliptic surfaces and K3 surfaces.

## Key findings

- First L-equivalence examples for genus one curves over non-closed fields
- L-equivalence for elliptic surfaces with multisection index five
- New L-equivalence cases for elliptic K3 surfaces of degree ten

## Abstract

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.