On relative Ulrich bundles and generalized Clifford algebras

TL;DR

This paper establishes a functorial equivalence between relatively Ulrich bundles on relative hypersurfaces and representations of generalized Clifford algebras, extending classical correspondences and revealing the complexity of Ulrich bundles.

Contribution

It generalizes the Ulrich-Clifford correspondence to a relative setting and introduces a purely algebraic framework for studying Ulrich bundles on hypersurfaces.

Findings

Relative hypersurfaces are Ulrich-wild with infinite families of indecomposable bundles.

Relative hyperplanes have minimal Ulrich complexity of one.

Homological obstructions for higher degrees require advanced tools like matrix factorizations.

Abstract

Let $X$ be a smooth projective scheme and $E$ a vector bundle on $X$. For a relative hypersurface $Y_f \subset \mathbb{P}(E)$ of degree $d$ defined by a global section $f$, we establish a functorial equivalence between the category of relatively Ulrich bundles on $Y_f$ and the category of representations of the associated generalized Clifford algebra $C_f$. This equivalence generalizes the classical Ulrich-Clifford correspondence of Coskun-Kulkarni-Mustopa and provides a purely algebraic framework that bypasses geometric obstructions in the relative setting. As a first application, we prove that relative hypersurfaces are Ulrich-wild: there exist families of indecomposable relatively Ulrich bundles $\{E_N\}$ with \[ \dim \mathrm{Ext}^1_{Y_f}(E_N, E_N) \to \infty \quad \text{as } N \to \infty. \] We further show that relative hyperplanes possess a minimal Ulrich complexity of one. Moving beyond degree one, we illustrate how unavoidable homological obstructions require complex machinery, such as matrix factorizations, equivalently generalized Clifford algebras, to find solutions.