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

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.

Categorical Torelli theorems: results and open problems

“…This is because Theorem 3.2 holds in this more general setting. On the other hand, when the genus is 1, the assumption 𝕂 = ℂ cannot be removed as there are nonisomorphic smooth projective curves of genus 1 with equivalent derived categories (see [4,139]).…”

Factorization centers in dimension two and the Grothendieck ring of varieties

“…It is currently unknown if zero-dimensional varieties can be nontrivially L-equivalent. The smallest-dimensional example of nontrivial L-equivalence is that of genus one curves over non-closed fields [33]. See [22,33] for some details about conjectural relationship between L-equivalence and derived equivalence, and the references therein for the currently known examples.…”

“…The smallest-dimensional example of nontrivial L-equivalence is that of genus one curves over non-closed fields [33]. See [22,33] for some details about conjectural relationship between L-equivalence and derived equivalence, and the references therein for the currently known examples. Note that the classes in the Grothendieck ring are insensitive to nonreduced structure, hence when studying L-equivalence we can always assume schemes to be reduced.…”