I'm looking at hypersurfaces on $\mathbb{C}\mathbb{P}^2$. That is, the zero set of an irreducible homogeneous polynomial $f(x_0, x_1, x_2) = 0$. This corresponds to a divisor, $D_f$, let's say which is just the trivial sum of 1 times the hypersurface. Every divisor has an associated line bundle $\mathcal{O}_{\mathbb{P}^2}(D_f)$. Now, since we're working in projective space, we know this must be $\mathcal{O}_{\mathbb{P}^2}(k)$ for some $k \in \mathbb{Z}$. I am trying to find out what $k$ should be. I have gone over a few different arguments and think that no matter what hypersurface, your corresponding line bundle will be the trivial bundle, $\mathcal{O}_{\mathbb{P}^2}(0)$. And then this should work over any $\mathbb{P}^n$ then as well.

So the question is, is this correct? Are there straightforward ways to see this? Is there any intuition as to why this would be true? It seems odd to me that I can take any hypersurface and get the same bundle (and the trivial one at that!).

Thanks for the help.

  • 1
    $\begingroup$ Answer: $k=\operatorname{deg} f$. $\endgroup$ – user64687 May 12 '15 at 19:21
  • $\begingroup$ Look at: Hartshorne's book Algebraic Geometry. Chapter II, Section 6, Proposition 6.4. $\endgroup$ – user99126 May 12 '15 at 19:32
  • $\begingroup$ Thank you @user99126, I looked it up and believe it. And that's what I originally thought but convinced myself it was trivial. $\endgroup$ – user46348 May 12 '15 at 19:44

One good reference for this material is section $1$, chapter $1$ of Principles of Algebraic Geometry by Griffiths and Harris. In particular, the following statement can be found on page $136$:

Thus if $D$ is any divisor such that $[D] = L$, there exists a meromorphic section $s$ of $L$ with $(s) = D$, and for any meromorphic section $s$ of $L$, $L = [(s)]$.

Here $[D]$ denotes the line bundle associated to the divisor $D$, and $(s)$ is the divisor associated to a meromorphic section $s$.

Let $f$ be a homogeneous polynomial of degree $k$ on $\mathbb{C}^{n+1}$, and let $X \subset \mathbb{CP}^n$ be the corresponding zero locus. Note that $f$ is holomorphic section of $\mathcal{O}(k)$ and $(f) = X$ so the line bundle associated to the divisor $X$ is $\mathcal{O}(k)$.


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