I have been reading Hartshorne. I am trying to understand a bit about a0mple, very ample line bundles, line bundles generated by global sections. I am trying to find these in case of the Projective space $\mathbb{P}^n_k$. Line bundles on $\mathbb{P}^n_k$ are of the form $O(d)$ where $d$ is an integer.

1) $O(1)$ is a very ample line bundle, so any $O(d)$ for $d>0$ is a very ample line bundle. Since a very ample line bundle has to have sections, $O(d)$ for $d<0$ cannot be very ample. Similarly the structure sheaf is not very ample.

2) A line bundle $L$ is ample if $L^{\otimes d}$ is very ample for some $d$. Hence again $O(d)$ for $d>0$ are only ample.

3) A globally generated line bundle has to have sections. So again only $O(d)$ for $d\geq 0$ are globally generated.

Am I right? Or is there some mistake?

  • $\begingroup$ yes, you are right. $\endgroup$ – Chen Jiang Jun 6 '16 at 8:28

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