The intersection of three quadrics in P^7 revisited

TL;DR

This work constructs and analyzes the noncommutative geometry associated to a very general intersection of three quadrics in $\mathbb{P}^7$ via twisted derived categories on the associated surface with a $2$-torsion Brauer class. By computing the twisted numerical $K$-lattice and comparing it to Mukai lattices of smooth surfaces, the authors prove that the twisted category is genuinely noncommutative and cannot be derived-equivalent to any smooth projective surface. They introduce and employ the framework of $\alpha$-twisted Euler pairings and $B$-twisted Mukai vectors, together with the lattice of Hodge data; these enable a new, lattice-theoretic obstruction to equivalence with commutative geometry. As a consequence, they obtain a new proof of the irrationality of a very general intersection of three quadrics in $\mathbb{P}^7$ via the theory of Hodge atoms, reinforcing the nonrationality results of Hassett–Pirutka–Tschinkel.

Abstract

We show that the natural nc-space attached to an intersection of three quadrics in P^7 is truly non-commutative. In particular, its associated numerical K-lattice is not isomorphic to the K-lattice of any smooth projective surface, so the relevant derived category is not equivalent to the derived category of any smooth projective surface. Using the new theory of Hodge atoms, this reproves the irrationality of a very general intersection of three quadrics in P7.

Theorems & Definitions (10)

• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Remark 2.4
• Lemma 3.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Corollary 4.4
• Theorem 4.5