On relative Ulrich bundles and generalized Clifford algebras

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.

29 Pages, All comments are welcome