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Entry loci and ranks

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Abstract

We study entry loci of varieties and their irreducibility from the perspective of X-ranks with respect to a projective variety X. These loci are the closures of the points that appear in an X-rank decomposition of a general point in the ambient space. We look at entry loci of low degree normal surfaces in \({\mathbb {P}}^4\) using Segre points of curves; the smooth case was classically studied by Franchetta. We introduce a class of varieties whose generic rank coincides with the one of its general entry locus, and show that any smooth and irreducible projective variety admits an embedding with this property.

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Acknowledgements

The first author is partially supported by MIUR and GNSAGA of INdAM (Italy). The second author is supported by the grant NWO Den Haag no. 38-573 of Jan Draisma. The authors would like to thank Francesco Russo for useful discussions and anonymous referees for valuable comments on the first version of this paper.

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Correspondence to Emanuele Ventura.

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Ballico, E., Ventura, E. Entry loci and ranks. manuscripta math. 165, 559–581 (2021). https://doi.org/10.1007/s00229-020-01216-z

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