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 1 The curve is said to be projectively normal if the map is an isomorphism.
For example, projective normality means that the map
is surjective; since it is injective (or would be contained in a hyperplane), it is an isomorphism. In particular, this means that the subspace 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 that defined the map 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 on for each , and natural linear series of given by the subspaces of given by the images from . Equivalently, one takes the linear series of divisors given by intersections of degree hypersurfaces in with . The projective normality condition is precisely that this linear series — defined by global degree 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 of genus , one has a canonical imbedding
and it is a theorem of Max Noether that this imbedding realizes as a projectively normal subvariety of . In other words, this says that, for example, the quadrics in cut out a complete linear series on . The quadrics in are precisely , and the content of the theorem is that
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 curves to the moduli stack of principally polarized -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 be normal (i.e., integrally closed). In fact, the ring is always normal: it is the global sections of the sheaf of normal domains, and is finite over (since they agree in high enough dimensions), so the integral closure of is .
2. Geometric and cohomological reformulations
Let’s work out a few consequences of projective normality. Let’s keep the notation of the previous section: is a degree , genus curve.
Choose a hyperplane , cut out by a section . Then is a zero-dimensional scheme, which implies that is regular along . In particular, it implies that the higher groups all vanish, or that the scheme-theoretic intersection cannot be “derived” any further (by taking derived tensor products).
In studying and its imbedding in projective space — for instance, in studying the hypersurfaces lies on — a basic tool is the formation of these types of hyperplane sections , which have the benefit of being simply configurations of points (or rather, zero-dimensional schemes) inside a smaller . For example, a hypersurface containing must restrict to a hypersurface in containing , and we can get information about hypersurfaces containing 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 . Then the curve is projectively normal if and only if every hypersurface in containing is the restriction of a hypersurface in containing .
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) is projectively normal if and only if, for every , we have
where is the ideal sheaf of .
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
obtained by twisting the exact sequence for by . Taking cohomology, and observing that of a line bundle on vanishes (unless ), we have an exact sequence
This exact sequence shows that the vanishing of is equivalent to the surjectivity of , which is projective normality. That proves the cohomological criterion.
Let’s go back and consider the more geometric statement. We have an exact sequence
which we can tensor with : the terms vanish because any hyperplane is cut out by an element regular with respect to . Doing this, and twisting by , we get an exact sequence
where is the ideal sheaf of inside . Once again, we’re using the vanishing of -terms to conclude that .
When we take global sections, we find that the obstruction to lifting a hypersurface in containing (that is, an element of ) to a hypersurface in containing is an element of . So if these ‘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
for each , and letting 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 , or that the linear series cut out by degree hypersurfaces on 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 hypersurfaces.
For example, we can always choose an imbedding of such that is linearly normal: that is, so that the linear series defining the imbedding is complete, or so that
In this case, the conclusion is that any quadric in the hyperplane section containing can be lifted to a quadric in containing . Any points in lie on a quadric (because there’s a -dimensional space of quadratic equations in ), so we can conclude that if , then lies on a quadric.
3. Nonspecial imbeddings
We’ll say that a line bundle on is nonspecial if . For instance, any line bundle of degree is nonspecial. Let’s consider the case where is imbedded so that is nonspecial. This is a very good case for several reasons.
Example 3 A nonspecially imbedded curve has the property that the Hilbert scheme of curves in is smooth at the point corresponding to . (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 . For our purposes, that means that obstructions to deforming live in the vector space , and that if these obstructions vanish, then the infinitesimal lifting property implies smoothness.
In fact, for a nonspecially imbedded curve, we have . For any curve, we have surjections
where the first surjection comes from the Euler sequence and the second surjection comes from . Since we are on a curve, we get a surjection in ,
and consequently on a nonspecially imbedded curve.
Example 4 It is a theorem of Halphen that any curve can be imbedded nonspecially into projective space via a degree line bundle once (and no better). In fact, a general divisor of degree 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
is a surjection, then one gets projective normality. To see this, let’s note that there is an exact sequence
and the long exact sequence in cohomology shows that we get a surjection:
Precomposing with the (by assumption) surjective map , we can conclude that
is a surjection. This is often phrased by saying that the points in impose independent conditions on quadrics in . In other words, if we pick one of the points, say , in — and let’s assume that is general enough so that is a transverse intersection now — then there’s a quadric in not passing through but passing through .
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 impose independent conditions on degree hypersurfaces for . If we look at the long exact sequence associated to
we find from these independent conditions that for and .
Using the exact sequence
we can conclude that, for , . Now returning to the exact sequence
we conclude that for , we have injections
and these are consequently all zero. Since we know that by linear normality, we’ve now completely proved projective normality.