Dimension of the global sections of the Serre twisting sheaf on a curve

Question

Let, $C$ be a projective curve (not necessarily reduced), $i:C \to \mathbb{P}^n$ be a closed immersion. Does there exist a bound on/geometric interpretation of the dimension of $H^0(i^*(\mathcal{O}_{\mathbb{P}^n}(1))$?

Answer 1

Thinking about $i^\ast \mathcal O_{\mathbb{P}^n}(1)$ as $\mathcal O_C(1)$, there is a natural exact sequence

$$0 \to I_C(1) \to \mathcal O_{\mathbb{P}^n}(1) \to \mathcal O_C(1) \to 0.$$

Assuming $C$ is non-degenerate, we obtain an exact sequence in cohomology

$$0\to H^0(\mathcal O_{\mathbb{P}^n}(1))\to H^0(\mathcal O_C(1))\to H^1( I_C(1))\to 0.$$

It folllows that $h^0(\mathcal O_C(1))\geq n+1$, with equality if and only if the $H^1$ group above vanishes. The vanishing of this $H^1$ is called linear normality of the embedded curve. There are several different properties which are equivalent to linear normality of an embedded curve, some of which are more or less geometric:

(1) Hyperplanes cut a complete linear series on $C$.

(2) $C$ is embedded by a complete linear series.

(3) $C$ is not a projection of a non-degenerate embedding from a higher-dimensional projective space.

(4) $H^1(I_C(1))=0$

In practice, the fourth property is usually the most useful for attempting to show a curve is linearly normal, as you can compute this cohomology group via other methods, such as resolving the ideal sheaf.