Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations
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| Formato: | Preprint |
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2024
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| _version_ | 1866915042436317184 |
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| author | Buczyńska, Weronika Buczyński, Jarosław Farnik, Łucja |
| author_facet | Buczyńska, Weronika Buczyński, Jarosław Farnik, Łucja |
| contents | For a fixed projective scheme X, a property P of line bundles is satisfied by sufficiently ample line bundles if there exists a line bundle L_0 on X such that P(L) holds for any L with (L - L_0) ample. As an example, sufficiently ample line bundles are very ample, moreover, for a normal variety X, the embedding corresponding to sufficiently ample line bundle is projectively normal. The grandfather of such properties and a basic ingredient used to study this concept is Fujita vanishing theorem, which is a strengthening of Serre vanishing to sufficiently ample line bundles. The r-th cactus variety of X is an analogue of secant variety and it is defined using linear spans of finite schemes of degree r. In this article we show that cactus varieties of sufficiently ample embeddings of X are set-theoretically defined by minors of matrices with linear entries. The topic is closely related to conjectures of Eisenbud-Koh-Stillman, which was proved by Ginensky in the case X a smooth curve. On the other hand Sidman-Smith proved that the ideal of sufficiently ample embedding of any projective scheme X is generated by 2 x 2 minors of a matrix with linear entries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_00709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations Buczyńska, Weronika Buczyński, Jarosław Farnik, Łucja Algebraic Geometry 14N07 (Primary) 14M12, 14F17, 14C05 (Secondary) For a fixed projective scheme X, a property P of line bundles is satisfied by sufficiently ample line bundles if there exists a line bundle L_0 on X such that P(L) holds for any L with (L - L_0) ample. As an example, sufficiently ample line bundles are very ample, moreover, for a normal variety X, the embedding corresponding to sufficiently ample line bundle is projectively normal. The grandfather of such properties and a basic ingredient used to study this concept is Fujita vanishing theorem, which is a strengthening of Serre vanishing to sufficiently ample line bundles. The r-th cactus variety of X is an analogue of secant variety and it is defined using linear spans of finite schemes of degree r. In this article we show that cactus varieties of sufficiently ample embeddings of X are set-theoretically defined by minors of matrices with linear entries. The topic is closely related to conjectures of Eisenbud-Koh-Stillman, which was proved by Ginensky in the case X a smooth curve. On the other hand Sidman-Smith proved that the ideal of sufficiently ample embedding of any projective scheme X is generated by 2 x 2 minors of a matrix with linear entries. |
| title | Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations |
| topic | Algebraic Geometry 14N07 (Primary) 14M12, 14F17, 14C05 (Secondary) |
| url | https://arxiv.org/abs/2412.00709 |