Injection dimensions of projective varieties

Abstract

We explore injective morphisms from complex projective varieties $X$ to projective spaces ${\mathbb{P}}^s$ of small dimension. Based on connectedness theorems, we prove that the ambient dimension $s$ needs to be at least $2\dim X$ 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 $X\hookrightarrow {\mathbb{P}}^n$ by linear projections. Our focus then lies on $X={\mathbb{P}}^{n_1}\times \dots \times {\mathbb{P}}^{n_r}$, in which case there is a close connection to secant loci of Segre--Veronese varieties and the rank $2$ geometry of partially symmetric tensors, as well as on $X=\mathbb{P}(q_0,\dots ,q_n)$, which is linked to separating invariants for representations of finite cyclic groups. We showcase three techniques for constructing injections $X\to {\mathbb{P}}^{2\dim X}$ in specific cases.

27 pages

References in corpus (1)

• The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)

Cited by in corpus (1)

• Algebraic compressed sensing