Nature of Invertible Sheaves in which there are no global sections. - MathOverflow

Nature of Invertible Sheaves in which there are no global sections.

2009-12-08T20:10:23Z

EDIT: Let me try to make the question clearer.

Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many properties, it is known that the space of global sections of such sheaves $V=\Gamma({\mathbb{P}^n},\mathcal{O}(d))$ encode information about embeddings between the projective spaces $\mathbb{P}^n$ and $\mathbb{P}(V)=\mathbb{P}^N$.

My question is the following. What kind of information do the sheaves $\mathcal{O}(d)$ encode when $d<0$?. By "information" I mean, in which context do they appear and seem to be helpful?

References are very welcome.

Answer by Alberto García-Raboso for Nature of Invertible Sheaves in which there are no global sections.

2009-12-08T20:46:17Z

I'm not sure what if this is what you are looking for, but here goes. All the information that you are associating to sheaves $\mathcal{O}(d)$ for positive $d$ seems to be essentially attached to their global sections. By Serre duality, $H^0(\mathcal{O}(-d)) \cong H^n(\mathcal{O}(d-n-1))$, and so these global sections talk about the other nonzero cohomology group of the invertible sheaves of positive degree.

Answer by Andrew Critch for Nature of Invertible Sheaves in which there are no global sections.

2009-12-08T23:45:07Z

They can still give you (non-canonical) rational maps to $\mathbb{P}^n$:

Even when an invertible sheaf $L$ on $X$ has no global sections, one can still find open subsets $U$ of $X$ such that $L|_U$ is globally generated, for example when $U$ is affine. This isn't so interesting, but if you can find such a $U$ that is not contained in an affine, then $L|_U$ might not be trivial, and then you might get morphisms from $U$ to $\mathbb{P}^n$ (by choosing generators) not coming from $\mathcal{O}_U$. If $X$ was integral, then $U$ will be automatically dense, so you get a rational map from $X$ to $\mathbb{P}^n$.

One way you could look for such a $U$ is to pick $n$ elements of the stalk $L_x$ at a point $x$, then intersect $n$ neighborhoods on which these elements extend, remove their common zero locus, and let the result be
$U$. If $U$ isn't contained in an affine, maybe you've found something cool. If you really wanted you could try working out a nice description for the rational map you've defined (though I've never done this). Even if $U$ was affine, maybe you've found a more interesting description of a less interesting map.

From a very different perspective, since $\mathcal{O}(-d)$ is the inverse of $\mathcal{O}(d)$, in some sense any "information" it contains is obtainable from inverting the transition functions of $\mathcal{O}(d)$... so with this restrictive view of "information", perhaps it would be interesting to study what sort of morphisms/maps to $\mathbb{P}^n$ one can get from an invertible sheaf $L$ on $X$ when neither $L$ nor $L^\vee$ is ample/very ample.

Answer by Ben Webster for Nature of Invertible Sheaves in which there are no global sections.

2009-12-09T00:24:22Z

You can still get a geometric realization of the Chern class: you look at a meromorphic section of your bundle, and you take minus the sum of the divisors on which it has a pole (weighted by order) plus the sum of the divisors where it has a 0 (also weighted by order).