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Let $\mathcal F$ be a coherent sheaf on $\mathbb P^r$, the $r$-dimensional projective space over an algebraically closed field $k$. The support of $\mathcal F$, namely $$\textrm{Supp }\mathcal F=\{x\in \mathbb P^r\,|\,\mathcal F_x\neq 0\}\subset \mathbb P^r,$$ carries a closed subscheme structure (given by the kernel of the sheaf homomorphism $\mathcal O_{\mathbb P^r}\to \mathcal End\,\mathcal F$). One can then look at the dimension of $\textrm{Supp }\mathcal F$, which lies between $0$ and $r$. (This is called the dimension of $\mathcal F$)

I was wondering whether there is some nice characterization of:

  1. Coherent sheaves $\mathcal F$ such that $\dim\,(\textrm{Supp }\mathcal F)=0$;

  2. Coherent sheaves $\mathcal F$ such that $\dim\,(\textrm{Supp }\mathcal F)=r$.

By "nice characterization" I just mean I would like to know if these sheaves are well understood, and one can describe them explicitly.

For instance, in 1, we find the structure sheaf of any $0$-dimensional subvariety of $\mathbb P^r$. I do not know whether there are others.

Thank you.

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For 1. recall that a $0$-dimensional closed subset of a noetherian scheme is finite and discrete. – Martin Brandenburg Sep 15 '13 at 19:48
@MartinBrandenburg: it seems to me that there only the structure sheaves of 0-dimensional subvarieties (not necessarily irreducible) and sheaves of ideas inside these. What am I missing in your comment? – Brenin Sep 17 '13 at 8:31

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