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.

For example, projective normality means that the map

$\displaystyle H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(1)) \rightarrow H^0( C, \mathcal{O}_C(1))$

is surjective; since it is injective (or ${C}$ would be contained in a hyperplane), it is an isomorphism. In particular, this means that the subspace ${V \subset H^0( C, \mathcal{O}_C(1))}$ must be the whole thing.

Let’s try to rephrase some of this in the language of linear systems. By definition, the previous paragraph stated that the linear system on ${C}$ that defined the map ${C \rightarrow \mathbb{P}^r}$ was a complete linear system: it contained all divisors in the appropriate linear equivalence class. But it is also saying more. One has line bundles ${\mathcal{L}^{n} = \mathcal{O}_C(n)}$ on ${C}$ for each ${n}$, and natural linear series of ${\mathcal{O}_C(n)}$ given by the subspaces of ${H^0( \mathcal{O}_C(n))}$ given by the images from ${H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(n))}$. Equivalently, one takes the linear series of divisors given by intersections of degree ${n}$ hypersurfaces in ${\mathbb{P}^r}$ with ${C}$. The projective normality condition is precisely that this linear series — defined by global degree ${n}$ hypersurfaces — is complete.

Example 1 A basic example (or family of examples) of a projectively normal curve is given by the canonical curves. Given a non-hyperelliptic curve ${C}$ of genus ${g}$, one has a canonical imbedding

$\displaystyle C \hookrightarrow \mathbb{P}^{g-1} = \mathbb{P}( H^0( \Omega_C)),$

and it is a theorem of Max Noether that this imbedding realizes ${C}$ as a projectively normal subvariety of ${\mathbb{P}^{g-1}}$. In other words, this says that, for example, the quadrics in ${\mathbb{P}^{g-1}}$ cut out a complete linear series on ${C}$. The quadrics in ${\mathbb{P}^{g-1}}$ are precisely ${\mathrm{Sym}^2 H^0( \Omega_C)}$, and the content of the theorem is that

$\displaystyle \mathrm{Sym}^2 H^0( \Omega_C) \rightarrow H^0( \Omega_C^2), \quad \mathrm{Sym}^3 H^0( \Omega_C) \rightarrow H^0( \Omega_C^3), \quad \dots ,$

are all surjective maps. Although I don’t really understand all this, Noether’s theorem is supposed to be the infinitesimal form of the Torelli theorem: that the map from the moduli stack of genus ${g}$ curves to the moduli stack of principally polarized ${g}$-dimensional abelian varieties, which sends a curve to its Jacobian, is an immersion away from the hyperelliptic locus.

The notion of “projective normality” can also be phrased in the following manner: the condition is that the homogeneous coordinate ring ${S}$ be normal (i.e., integrally closed). In fact, the ring ${\widetilde{S}}$ is always normal: it is the global sections of the sheaf ${\bigoplus_{n=0}^\infty \mathcal{O}_C(n)}$ of normal domains, and ${\widetilde{S}}$ is finite over ${S}$ (since they agree in high enough dimensions), so the integral closure of ${S}$ is ${\widetilde{S}}$.

2. Geometric and cohomological reformulations

Let’s work out a few consequences of projective normality. Let’s keep the notation of the previous section: ${C \subset \mathbb{P}^r}$ is a degree ${d}$, genus ${g}$ curve.

Choose a hyperplane ${H \subset \mathbb{P}^r}$, cut out by a section ${s \in H^0( \mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(1))}$. Then ${H \cap C}$ is a zero-dimensional scheme, which implies that ${s}$ is regular along ${C}$. In particular, it implies that the higher ${\mathrm{Tor}}$ groups ${\mathrm{Tor}_i( \mathcal{O}_H, \mathcal{O}_C)}$ all vanish, or that the scheme-theoretic intersection ${H \cap C}$ cannot be “derived” any further (by taking derived tensor products).

In studying ${C}$ and its imbedding in projective space — for instance, in studying the hypersurfaces ${C}$ lies on — a basic tool is the formation of these types of hyperplane sections ${H \cap C}$, which have the benefit of being simply configurations of points (or rather, zero-dimensional schemes) inside a smaller ${\mathbb{P}^{r-1}}$. For example, a hypersurface containing ${C}$ must restrict to a hypersurface in ${\mathbb{P}^{r-1} \simeq H}$ containing ${H \cap C}$, and we can get information about hypersurfaces containing ${C}$ by studying hypersurfaces containing zero-dimensional schemes. One of the useful properties of projectively normal curves is that we can go in the other direction.

Proposition 2 (Geometric criterion) Fix a hyperplane ${H \subset \mathbb{P}^r}$. Then the curve ${C \subset \mathbb{P}^r}$ is projectively normal if and only if every hypersurface in ${H}$ containing ${H \cap C}$ is the restriction of a hypersurface in ${\mathbb{P}^r}$ containing ${C}$.

Note in particular that the statement only requires that we check something for one hyperplane, but then we can go back and conclude the same for all hyperplanes. We get in particular a very geometric criterion for projective normality.

There is also a cohomological criterion, as follows:

Proposition 3 (Cohomological criterion) ${C \subset \mathbb{P}^r}$ is projectively normal if and only if, for every ${k \geq 1}$, we have

$\displaystyle H^1( \mathcal{I}_C(k)) = 0,$

where ${\mathcal{I}_C}$ is the ideal sheaf of ${C \subset \mathbb{P}^r}$.

Let’s prove the equivalence of these two criteria with the previous definition of projective normality, and we’ll start with the cohomological one. First, note that there is an exact sequence

$\displaystyle 0 \rightarrow \mathcal{I}_C(k) \rightarrow \mathcal{O}_{\mathbb{P}^r}(k) \rightarrow \mathcal{O}_C(k) \rightarrow 0,$

obtained by twisting the exact sequence for ${\mathcal{O}_C \simeq \mathcal{O}_{\mathbb{P}^r}/\mathcal{I}_C}$ by ${k}$. Taking cohomology, and observing that ${H^1}$ of a line bundle on ${\mathbb{P}^r}$ vanishes (unless ${r = 1}$), we have an exact sequence

$\displaystyle 0 \rightarrow H^0( \mathcal{I}_C(k)) \rightarrow H^0(\mathcal{O}_{\mathbb{P}^r}(k)) \rightarrow H^0(\mathcal{O}_C(k)) \rightarrow H^1(\mathcal{I}_C(k)) \rightarrow 0.$

This exact sequence shows that the vanishing of ${H^1(\mathcal{I}_C(k))}$ is equivalent to the surjectivity of ${H^0(\mathcal{O}_{\mathbb{P}^r}(k)) \rightarrow H^0(\mathcal{O}_C(k)) }$, which is projective normality. That proves the cohomological criterion.

Let’s go back and consider the more geometric statement. We have an exact sequence

$\displaystyle 0 \rightarrow \mathcal{O}_{\mathbb{P}^r}(-1) \rightarrow \mathcal{O}_{\mathbb{P}^r} \rightarrow \mathcal{O}_H \rightarrow 0,$

which we can tensor with ${\mathcal{I}_C}$: the ${\mathrm{Tor}_1}$ terms vanish because any hyperplane is cut out by an element regular with respect to ${\mathcal{O}_C}$. Doing this, and twisting by ${k}$, we get an exact sequence

$\displaystyle 0 \rightarrow \mathcal{I}_C(k-1) \rightarrow \mathcal{I}_C(k) \rightarrow \mathcal{I}_{C \cap H}^H(k) \rightarrow 0,$

where ${\mathcal{I}_{C \cap H}^H}$ is the ideal sheaf of ${C \cap H}$ inside ${H}$. Once again, we’re using the vanishing of ${\mathrm{Tor}_1}$-terms to conclude that ${\mathcal{I}_{C \cap H}^H \simeq \mathcal{I}_C \otimes \mathcal{O}_H}$.

When we take global sections, we find that the obstruction to lifting a hypersurface in ${H}$ containing ${C \cap H}$ (that is, an element of ${H^0( \mathcal{I}_{C \cap H}^H(k))}$) to a hypersurface in ${\mathbb{P}^r}$ containing ${C}$ is an element of ${H^1( \mathcal{I}_C(k-1))}$. So if these ${H^1}$‘s vanish, there is no obstruction and the condition of the geometric criterion holds. Conversely, if the condition of the geometric criterion holds, the long exact sequence in cohomology shows that we have injections

$\displaystyle H^1( \mathcal{I}_C(k-1)) \hookrightarrow H^1( \mathcal{I}_C(k)),$

for each ${k \geq 1}$, and letting ${k \rightarrow \infty}$ so that these groups vanish, we find that the cohomological criterion is satisfied. That completes the proof of the geometric criterion.

Example 2 The above analysis actually showed a little extra. If we knew that ${H^1( \mathcal{I}_C(k-1)) = 0}$, or that the linear series cut out by degree ${k-1}$ hypersurfaces on ${C}$ was complete (see the proof of the cohomological criterion), then we could conclude that the conclusion of the geometric criterion held, but only for degree ${k}$ hypersurfaces.

For example, we can always choose an imbedding of ${C}$ such that ${C}$ is linearly normal: that is, so that the linear series defining the imbedding is complete, or so that

$\displaystyle H^0( \mathcal{O}_{\mathbb{P}^r}(1)) \simeq H^0( \mathcal{O}_C(1)).$

In this case, the conclusion is that any quadric in the hyperplane section ${H \subset \mathbb{P}^r}$ containing ${H \cap C}$ can be lifted to a quadric in ${\mathbb{P}^r}$ containing ${C}$. Any ${\binom{r+1}{2} -1 }$ points in ${H}$ lie on a quadric (because there’s a ${\binom{r+1}{2}}$-dimensional space of quadratic equations in ${H}$), so we can conclude that if ${d < \binom{r + 1}{2}}$, then ${C}$ lies on a quadric.

3. Nonspecial imbeddings

We’ll say that a line bundle ${\mathcal{L}}$ on ${C}$ is nonspecial if ${H^1(\mathcal{L}) \neq 0}$. For instance, any line bundle of degree ${\geq 2g-1}$ is nonspecial. Let’s consider the case where ${C \subset \mathbb{P}^r}$ is imbedded so that ${\mathcal{O}_C(1)}$ is nonspecial. This is a very good case for several reasons.

Example 3 A nonspecially imbedded curve ${C \subset \mathbb{P}^r}$ has the property that the Hilbert scheme of curves in ${\mathbb{P}^r}$ is smooth at the point corresponding to ${C}$. (In general, it is known that a sort of “Murphy’s law” holds for Hilbert schemes of smooth curves in projective space: all sorts of terrible singularities occur.) To see this, we’ll use the fact that the Hilbert scheme has a well-behaved tangent-obstruction theory with values in the cohomology of the normal bundle ${\mathcal{N}_C}$. For our purposes, that means that obstructions to deforming ${C}$ live in the vector space ${H^1( \mathcal{N}_C)}$, and that if these obstructions vanish, then the infinitesimal lifting property implies smoothness.

In fact, for a nonspecially imbedded curve, we have ${H^1( \mathcal{N}_C) = 0}$. For any curve, we have surjections

$\displaystyle \mathcal{O}_{C}(1)^{r+1} \twoheadrightarrow T_{\mathbb{P}^r}|_C \twoheadrightarrow \mathcal{N}_C,$

where the first surjection comes from the Euler sequence and the second surjection comes from ${\mathcal{N}_C \simeq T_{\mathbb{P}^r}|C/T_C}$. Since we are on a curve, we get a surjection in ${H^1}$,

$\displaystyle H^1(\mathcal{O}_C(1))^{r+1} \twoheadrightarrow H^1(\mathcal{N}_C),$

and consequently ${H^1(\mathcal{N}_C) = 0}$ on a nonspecially imbedded curve.

Example 4 It is a theorem of Halphen that any curve ${C}$ can be imbedded nonspecially into projective space via a degree ${d}$ line bundle once ${d \geq g + 3}$ (and no better). In fact, a general divisor of degree $g + 3$ is very ample.

For a nonspecially imbedded (and linearly normal) curve, the claim is that projective normality is purely a condition at level 2. That is, once

$\displaystyle H^0(\mathcal{O}_{\mathbb{P}^r}(2)) \twoheadrightarrow H^0( \mathcal{O}_C(2))$

is a surjection, then one gets projective normality. To see this, let’s note that there is an exact sequence

$\displaystyle 0 \rightarrow \mathcal{O}_C(1) \rightarrow \mathcal{O}_C(2) \rightarrow \mathcal{O}_{C \cap H}(2) \rightarrow 0,$

and the long exact sequence in cohomology shows that we get a surjection:

$\displaystyle H^0(\mathcal{O}_C(2) ) \twoheadrightarrow H^0( \mathcal{O}_{C \cap H}(2)).$

Precomposing with the (by assumption) surjective map ${H^0(\mathcal{O}_{\mathbb{P}^r}(2)) \twoheadrightarrow H^0( \mathcal{O}_C(2)) }$, we can conclude that

$\displaystyle H^0(\mathcal{O}_{\mathbb{P}^r}(2)) \twoheadrightarrow H^0( \mathcal{O}_{C \cap H}(2))$

is a surjection. This is often phrased by saying that the points in ${C \cap H}$ impose independent conditions on quadrics in ${\mathbb{P}^r}$. In other words, if we pick one of the points, say ${p}$, in ${C \cap H}$ — and let’s assume that ${H}$ is general enough so that ${H \cap C}$ is a transverse intersection now — then there’s a quadric in ${\mathbb{P}^r}$not passing through ${p}$ but passing through ${C \cap H \setminus \left\{p\right\}}$.

Clearly, if we have that for quadrics, we have that for cubics, quartics, and so forth—we can throw in an extra hyperplane if we need to. So, more generally, the points of ${C \cap H}$ impose independent conditions on degree ${k}$ hypersurfaces for ${k \geq 2}$. If we look at the long exact sequence associated to

$\displaystyle 0 \rightarrow \mathcal{I}_{C \cap H}^{\mathbb{P}^r}(k) \rightarrow \mathcal{O}_{\mathbb{P}^r}(k) \rightarrow \mathcal{O}_{C \cap H}(k) \rightarrow 0,$

we find from these independent conditions that ${H^i( \mathcal{I}_{C \cap H}^{\mathbb{P}^r}(k)) =0}$ for ${k \geq 2}$ and ${i \geq 1}$.

Using the exact sequence

$\displaystyle 0 \rightarrow \mathcal{I}_{C \cap H}^{\mathbb{P}^r}(k-1) \rightarrow \mathcal{I}_{C \cap H}^{\mathbb{P}^r}(k) \rightarrow \mathcal{I}_{C \cap H}^H(k) \rightarrow 0,$

we can conclude that, for ${k \geq 3}$, ${H^1( \mathcal{I}_{C \cap H}^H(k)) = 0}$. Now returning to the exact sequence

$\displaystyle 0 \rightarrow \mathcal{I}_{C}(k-1) \rightarrow \mathcal{I}_C(k) \rightarrow \mathcal{I}_{C \cap H}^H(k) \rightarrow 0,$

we conclude that for ${k \geq 3}$, we have injections

$\displaystyle H^1(\mathcal{I}_{C}(k-1)) \rightarrow H^1(\mathcal{I}_C(k)) ,$

and these are consequently all zero. Since we know that ${H^1( \mathcal{I}_C(1)) = 0}$ by linear normality, we’ve now completely proved projective normality.