# Ample vector bundles and embeddings

If $$X$$ is a complete variety over a field, a line bundle $$L$$ is said to be very ample if there is a closed immersion from $$X$$ into a projective space, such that the pullback of $$\mathcal{O}(1)$$ is isomorphic to $$L$$. Consequently, an ample line bundle can be defined as a line bundle such that some tensor power of it is very ample.

There is also a notion of amplitude for vector bundles, which seems to have mostly been developed by Hartshorne. One definition that can be taken is that if $$E$$ is a vector bundle, then $$E$$ is ample if $$\mathcal{O}(1)$$ on $$\mathbb{P}(E)$$ is ample. This is the definition taken in Positivity in Algebraic Geometry II, by Robert Lazarsfeld, and in Example 6.1.6, he states that the tautological bundle on some Grassmanians is nef but not ample, and hence any provisional definition of ample vector bundles in terms of embeddings into Grassmanians is destined to fail.

My questions are the following:

1. What might be some conditions on $$E$$ so that it does give some embedding into a Grassmanian? Can this be geometrically realized as in the case of ample vector bundles, i.e., is there a notion of very ample vector bundle (which hopefully would include ample vector bundles) which do give embeddings?

2. Given any coherent sheaf $$F$$ over $$X$$, we can form its symmetric algebra $$S(F)$$, and then form $$\tilde X = \underline{\operatorname{Proj}}(S(E))$$. Could then one define an ample coherent sheaf as above, where $$\mathcal{O}(1)$$ on $$\tilde X$$ is ample? Could one also generalize this to a notion of embeddings, i.e., very ample coherent sheaves?

3. If the answer to the above is no; what about the special case of divisorial sheaves (reflexive, generically rank one)? Specifically I am interested in some notion which generalizes the canonical model of a smooth variety to that of Cohen-Macaulay varieties by somehow defining amplitude (in terms of embeddings) of the canonical (generalized) divisor $$\omega_X=h^{-n}(\omega_{\overline X}^\bullet)|_X=\mathscr Ext^{N-n}_{\mathbb P^N}(\mathscr O_{\overline X},\omega_{\mathbb P^N})|_X$$, where $$X \subset \mathbb{P}^N$$ is of dimension $$n$$.