Let be a (smooth) curve in projective space of some degree . We will assume that is *nondegenerate:* that is, that is not contained in a hyperplane. In other words, one has an abstract algebraic curve , and the data of a line bundle of degree on , and a subspace of dimension such that the sections in have no common zeros in .

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 inside . The curve is defined by a homogeneous ideal (consisting of all homogeneous polynomials whose vanishing locus contains ). The **homogeneous coordinate ring** of is defined via

it is an integral domain. Equivalently, it can be defined as the image of in . But that in turn suggests another natural ring associated to , which only depends on the line bundle and not the projective imbedding: that is the ring

where the multiplication comes from the natural maps for line bundles on . One has a natural map

which is injective by construction. Moreover, since higher cohomology always vanishes after enough twisting, the map is surjective in all large dimensions.

Definition 1The curve is said to beprojectively normalif the map is an isomorphism. (more…)