Derived category of the spinor 15-fold

TL;DR

This paper constructs a full exceptional Lefschetz collection on the spinor 15-fold, confirming a conjecture and suggesting potential extensions to other varieties.

Contribution

It provides the first explicit construction of a Lefschetz collection on the spinor 15-fold, advancing understanding of derived categories in algebraic geometry.

Findings

Confirmed a conjecture of Kuznetsov and Smirnov

Constructed a Lefschetz collection with specific twists and blocks

Speculated on similar collections for the Freudenthal E7-variety

Abstract

We construct a full exceptional Lefschetz collection on the spinor 15-fold consisting of a connected component of the space of orthogonal 6-dimensional subspaces of a 12-dimensional complex vector space, isotropic with respect of a fixed non-degenerate quadratic form. The collection is made of 2 twists of a 4-item block and 8 twists of a 3-item block, confirming a conjecture of Kuznetsov and Smirnov. We speculate that a similar collection might work for the Freudenthal E7-variety.