On relative Ulrich bundles and generalized Clifford algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913044659961856 |
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| author | Mondal, Soham Mukherjee, Anindya |
| author_facet | Mondal, Soham Mukherjee, Anindya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01611 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On relative Ulrich bundles and generalized Clifford algebras Mondal, Soham Mukherjee, Anindya Algebraic Geometry Commutative Algebra Representation Theory 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. |
| title | On relative Ulrich bundles and generalized Clifford algebras |
| topic | Algebraic Geometry Commutative Algebra Representation Theory |
| url | https://arxiv.org/abs/2604.01611 |