Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

TL;DR

This paper introduces new algebraic and categorical methods to analyze fibrations with fibers as complete intersections of quadrics, linking derived categories and Clifford algebras to rationality problems in algebraic geometry.

Contribution

It develops a theory of relative homological projective duality and Morita invariance of Clifford algebras, applying these to study derived categories and rationality of specific fibrations.

Findings

Established new categorical tools for analyzing complete intersections of quadrics.

Connected rationality questions to categorical representability in specific geometric cases.

Applied methods to fibrations with fibers of genus 1, Del Pezzo surfaces, and Fano threefolds.

Abstract

Let X -> Y be a fibration whose fibers are complete intersections of two quadrics. We develop new categorical and algebraic tools---a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting---to study semiorthogonal decompositions of the bounded derived category of X. Together with new results in the theory of quadratic forms, we apply these tools in the case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if Y is the projective line over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.