Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations
| Autor: | Buczyńska, Weronika, Buczyński, Jarosław, Farnik, Łucja |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 32 pages, 1 figure, Dedicated in memory of Gianfranco Casnati |
| Databáze: | arXiv |
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