Singular loci in varieties of tensors
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916564948746240 |
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| author | Chiu, Christopher Danelon, Alessandro Draisma, Jan |
| author_facet | Chiu, Christopher Danelon, Alessandro Draisma, Jan |
| contents | A Vec-variety is a suitable functor from finite-dimensional vector spaces to finite-dimensional varieties. Most varieties in the geometry of tensors, e.g. the variety of d-way tensors of slice rank at most r, are of this form. We prove that the singular locus of a Vec-variety is a proper closed Vec-subvariety, analogously to the situation for ordinary finite-dimensional varieties. Via earlier work of the third author, this implies that these singular loci admit a description by finitely many polynomial equations.
A natural follow-up question to our main result is whether a Vec-variety also admits a suitably functorial resolution of singularities. We establish some preliminary results in this direction in the regime where the dimension of evaluations of a Vec-variety grows linearly with that of the input vector space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular loci in varieties of tensors Chiu, Christopher Danelon, Alessandro Draisma, Jan Algebraic Geometry Representation Theory 14B05, 14L30, 14N07, 20G05 A Vec-variety is a suitable functor from finite-dimensional vector spaces to finite-dimensional varieties. Most varieties in the geometry of tensors, e.g. the variety of d-way tensors of slice rank at most r, are of this form. We prove that the singular locus of a Vec-variety is a proper closed Vec-subvariety, analogously to the situation for ordinary finite-dimensional varieties. Via earlier work of the third author, this implies that these singular loci admit a description by finitely many polynomial equations. A natural follow-up question to our main result is whether a Vec-variety also admits a suitably functorial resolution of singularities. We establish some preliminary results in this direction in the regime where the dimension of evaluations of a Vec-variety grows linearly with that of the input vector space. |
| title | Singular loci in varieties of tensors |
| topic | Algebraic Geometry Representation Theory 14B05, 14L30, 14N07, 20G05 |
| url | https://arxiv.org/abs/2501.07497 |