Let ${C \subset \mathbb{P}^r}$ be a (smooth) curve in projective space of some degree ${d}$. We will assume that ${C}$ is nondegenerate: that is, that ${C}$ is not contained in a hyperplane. In other words, one has an abstract algebraic curve ${C}$, and the data of a line bundle ${\mathcal{L} = \mathcal{O}_C(1)}$ of degree ${d}$ on ${C}$, and a subspace ${V \subset H^0( \mathcal{L})}$ of dimension ${r+1}$ such that the sections in ${V}$ have no common zeros in ${C}$.

In this post, I’d like to discuss a useful condition on such an imbedding, and some of the geometry that it leads to. Most of this material is, once again, from ACGH’s book Geometry of algebraic curves.

1. Projective normality

In general, there are two natural commutative graded rings one can associate to this data. First, one has the homogeneous coordinate ring of ${C}$ inside ${\mathbb{P}^r}$. The curve ${C \subset \mathbb{P}^r}$ is defined by a homogeneous ideal ${I \subset k[x_0, \dots, x_r]}$ (consisting of all homogeneous polynomials whose vanishing locus contains ${C}$). The homogeneous coordinate ring of ${C}$ is defined via $\displaystyle S = k[x_0, \dots, x_r]/I;$

it is an integral domain. Equivalently, it can be defined as the image of ${k[x_0, \dots, x_r] = \bigoplus_{n = 0}^\infty H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(n))}$ in ${\bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n))}$. But that in turn suggests another natural ring associated to ${C}$, which only depends on the line bundle ${\mathcal{L}}$ and not the projective imbedding: that is the ring $\displaystyle \widetilde{S} = \bigoplus_{n = 0}^\infty H^0( C, \mathcal{O}_C(n)),$

where the multiplication comes from the natural maps ${H^0(\mathcal{M}) \otimes H^0(\mathcal{N}) \rightarrow H^0( \mathcal{M} \otimes \mathcal{N})}$ for line bundles ${\mathcal{M}, \mathcal{N}}$ on ${C}$. One has a natural map $\displaystyle S \hookrightarrow \widetilde{S},$

which is injective by construction. Moreover, since higher cohomology always vanishes after enough twisting, the map ${S \rightarrow \widetilde{S}}$ is surjective in all large dimensions.

Definition 1 The curve ${C \subset \mathbb{P}^r}$ is said to be projectively normal if the map ${S \hookrightarrow \widetilde{S}}$ is an isomorphism. (more…)