Injection dimensions of projective varieties

Paul Görlach

Contact the author: Please use for correspondence this email.
Submission date: 28. May. 2019
Bibtex
MSC-Numbers: 14N05, 51N35, 13A50
Keywords and phrases: injection dimension, separating invariants, rank two geometry
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
We explore injective morphisms from complex projective varieties X to projective spaces ℙ^s of small dimension. Based on connectedness theorems, we prove that the ambient dimension s needs to be at least 2dimX for all injections given by a linear subsystem of a strict power of a line bundle. Using this, we give an example where the smallest ambient dimension cannot be attained from any embedding X ⊆ ℙⁿ by linear projections. Our focus then lies on X = ℙ^n₁ ×…× ℙ^n_r, in which case there is a close connection to secant loci of SegreVeronese varieties and the rank 2 geometry of partially symmetric tensors, as well as on X = ℙ(q₀,…,q_n), which is linked to separating invariants for representations of finite cyclic groups. We showcase three techniques for constructing injections X → ℙ^{2 dim X} in specific cases.