TY - JOUR

T1 - L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

AU - Shinder, Evgeny

AU - Zhang, Ziyu

N1 - Funding Information: We would like to thank Tom Bridgeland, Tom Fisher, Sergey Galkin, Daniel Huybrechts, Alexander Kuznetsov, Jayanta Manoharmayum, C.S. Rajan, Matthias Schütt, Constantin Shramov and Damiano Testa for helpful discussions and e-mail correspondences. In particular, we thank Alexander Kuznetsov for explaining to us how to prove the duality in Proposition 2.8, and for his comments on a draft of the paper. We thank the University of Sheffield, Leibniz University Hannover and the University of Bonn for giving us opportunities to travel and collaborate, and the Max-Planck-Institut für Mathematik in Bonn for the excellent and inspiring working conditions, where much of this work has been written. We also thank the referees for carefully reading the manuscript and their suggestions.

PY - 2020/4/23

Y1 - 2020/4/23

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85083492050&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1907.01335

DO - 10.48550/arXiv.1907.01335

M3 - Article

AN - SCOPUS:85083492050

VL - 52

SP - 395

EP - 409

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 2

ER -