Injection dimensions of projective varieties
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Submission date: 28. May. 2019
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.
We explore injective morphisms from complex projective varieties X to projective spaces ℙs 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 = ℙn1 ×…× ℙnr, 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,…,qn), which is linked to separating invariants for representations of ﬁnite cyclic groups. We showcase three techniques for constructing injections X → ℙ2 dim X in speciﬁc cases.