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

Ziyu Zhang,Evgeny Shinder

doi:10.1112/blms.12339

Ziyu Zhang, Evgeny Shinder

Open Access

https://doi.org/10.1112/blms.12339

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Abstract

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.

Highlights

In this paper we study L-equivalence for genus one curves and elliptic surfaces, in particular for elliptic K3 surfaces
We show that (1.1) holds for X and Y when n 4. This is the first existing construction of nontrivial L-equivalence for curves, as all the previous constructions were for K3 surfaces or Calabi-Yau varieties of higher dimension
As it is often the case with proving L-equivalence we relate the geometry of X and Y to Homological Projective Duality of A

Summary

Introduction

(see Theorem 2.9) If X is a genus one curve with a line bundle of degree 5 and Y = Jac2(X), X and Y are L-equivalent, and in general this L-equivalence is nontrivial. This is the first existing construction of nontrivial L-equivalence for curves, as all the previous constructions were for K3 surfaces or Calabi-Yau varieties of higher dimension As it is often the case with proving L-equivalence we relate the geometry of X and Y to Homological Projective Duality of A. These explicit descriptions show in particular that for d 5 the corresponding moduli spaces are rational It is this description in the d = 5 case, together with the geometric characterization of the self-map Jac on the moduli space of degree 5 genus one curves, given in Proposition 1.4 that allows us to prove L-equivalence. This is a genuinely new instance of nontrivial L-equivalence between K3 surfaces

Dual elliptic quintics

Elliptic surfaces of index five