Very ample line bundle and the induced embedding

Question

Over $\mathbb C$, let $X$ be a projective variety and let $\mathcal L$ be a very ample line bundle. Then there is an induced embedding $X\to \mathbb P(V)$ for $V=H^0(X;\mathcal L)^*$.

It is easy to see there is a surjection $\Gamma(\mathcal O(1))\to\Gamma(\mathcal L)$. I wonder if this is also true for $\mathcal O(n)$ and $\mathcal L^n$ in general.

Or equivalently can people find a counterexample of a projective variety with a very ample line bundle such that $\bigoplus H^0(X;L^n)$ is not generated by degree one elements?

Answer 1

If $X$ is normal, the condition that $H^0(\Bbb P^n,\mathcal{O}(d))\to H^0(X,\mathcal{O}_X(d))$ is surjective for all $d\geq 0$ is equivalent to $X\subset\Bbb P^n$ being projectively normal. (The condition that $\Gamma(\mathcal{O}(1))\to\Gamma(\mathcal{L})$ is surjective is called being linearly normal.) There's essentially no reason for a linearly normal variety to be projectively normal. For instance, here's an example from MO by user abx:

Here is a concrete example : take a curve $C$ of genus 4 and a line bundle $L$ on $C$ of degree 7 which is very ample (that means that $L$ is not of the form $K_C(p+q-r)$ for $p,q,r\in C$). Then $L$ embeds $C$ in $\mathbb{P}^3$, $\dim H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(2))=10$ and $\dim H^0(C,L^2)= 11$, so the above map cannot be surjective.