Weak Kodaira Vanishing - Hartshorne III.7.1

Question

In the Serre Duality section of Algebraic Geometry by Robin Hartshorne, the following exercise is posed: If $X$ is an integral projective scheme over a field $k$, prove that an ample invertible sheaf $\mathcal L$ is such that $\mathcal L^{-1}$ has no nonzero global sections. This is fairly straightforward when there are additional hypotheses on $X$ (for example, those that let you use Theorem 7.6b), but in the stated generality I'm not sure what to do. I can reduce to the case of $\mathcal L$ very ample, but that's about as far as I get.

Answer 1

You can show that, for $n\gg0$, $\mathcal L^n$ has a lot of global sections. ($\dim_k(H^0(X,\mathcal L^n))\geq \dim X+1$).

If $\mathcal L^{-1}$ has a nonzero global section, then so does $\mathcal L^{-n}$. Tensor these with those of $\mathcal L^n$ to get lots of global sections of $\mathcal{O}_X$. But we know there is only one global section (up to scalar of course).

Answer 2

Similar with the above answer. (But I cannot add a comment?)

Since $\mathscr{L}$ is ample, we just pick an $n$ such that $\mathscr{L}^{n}$ is very ample, and $\dim_{k}(H^{0}(X,\mathscr{L}^{n}))=\dim X+1$. If $\mathscr{L}^{-1}$ has a nonzero global section $s$, the morphism $H^{0}(X,\mathscr{L}^{n})\rightarrow H^{0}(X,\mathscr{O}_{X})$ as $\times s^{n}$ has image more than dimension one, which makes a contradiction.