Clifford spaces of empty intersections of quadrics

TL;DR

This paper studies Clifford spaces derived from empty intersections of quadrics, revealing their derived categories, dualities, and algebraic structures, and introduces examples of noncommutative projective spaces.

Contribution

It demonstrates that Clifford spaces have derived categories generated by full exceptional collections, and explores their dualities and algebraic deformations in special cases.

Findings

Derived category of Clifford space is generated by a full exceptional collection.

Maximal Clifford space is homologically projectively dual to the second Veronese embedding.

Minimal Clifford space's algebra is a Koszul flat deformation of a polynomial algebra.

Abstract

Given a linear space $U \subset \mathrm{Sym}^2V^\vee$ of quadrics in a projective space $\mathbb{P}(V)$ whose intersection is empty, we consider the corresponding Clifford space -- the projective space $\mathbb{P}(U)$ endowed with the even part of Clifford algebras as a sheaf of algebras. We show that the derived category of a Clifford space is generated by a full exceptional collection that extends to a 1-periodic helix and the Clifford space is equivalent to the noncommutative projective spectrum of the corresponding graded algebra. We discuss two special cases of Clifford spaces in more detail. The first is the maximal Clifford space, associated to the complete linear system $U = \mathrm{Sym}^2V^\vee$ of quadrics. It is homologically projectively dual to the second Veronese embedding of the projective space $\mathbb{P}(V)$. We show that the corresponding graded algebra is the maximal multiplicity-free direct sum of all polynomial representations of $\mathrm{GL}(V)$ and describe its dual $\mathrm{A}_\infty$-algebra. The second is a minimal Clifford space, associated to a linear system of quadrics with $\dim(U) = \dim(V)$. We show that the corresponding graded algebra is a Koszul flat deformation of a polynomial algebra and its dual algebra is a Frobenius flat deformation of an exterior algebra. In particular, a minimal Clifford space is an example of a noncommutative projective space.