RIEMANN-ROCH FOR CURVES

A smooth projective curve C over a field k is by definition a smooth irreducible projective variety of
dimension 1 over k. To begin with let us think of C ⊂ ℙ^{n} for some unspecified n. Let us begin with a more
detailed study of the arithmetic genus.

As an example consider a curve C ⊂ ℙ^{2} defined as the locus of vanishing of a homogeneous polynomial
F(X,Y,Z) of degree d. It is clear that the homogeneous co-ordinate ring of C is R = k[X,Y,Z]∕(F) and so
the dimension of R_{m} for all sufficiently large m is

Thus we obtain that the arithmetic genus of a curve of degree d in ℙ^{2} is (d - 1)(d - 2)∕2. We notice that
the dimension of R_{m} for small m does not obey this rule. In particular, the dimension of R_{1} is 3 as
soon as F has no linear factor. Understanding this phenomenon will lead us to “special” linear
series.

As mentioned in an earlier lecture, all morphisms from a curve C ⊂ ℙ^{n} can be seen as a composition
of the Veronese embeddings and projections. From the definition given above it is clear that
the Veronese embedding does not change the arithmetic genus. What about projection? By
iteration, it is enough to examine what happens when we project from a single point on the
curve. If we prove that the arithmetic genus is unchanged under an isomorphism induced by
such a projection, then we will have proved that the arithmetic genus of a curve is an intrinsic
invariant.

By a linear change of co-ordinates we can assume that point P we are projecting from has co-ordinates
(1 : 0 : : 0). In that case the projection is seen ring-theoretically as taking the sub-ring S of R generated
by the variables X_{1},…,X_{n}. Now, we are also assuming that the curve in ℙ^{n-1} is isomorphic to our original
curve C. If we examine what this means in terms of local co-ordinates on the curve C at the point P we see
that S_{m} consists of those F in R_{m} such that the first m Taylor coefficients of F∕X_{0}^{m} vanish at
the point. Now, we use the smooth-ness of the curve at the point which implies that there is
a linear polynomial (say X_{1}) such that X_{1}∕X_{0} is the local co-ordinate at the point P; then
X_{0}^{k}X_{1}^{m-k} are linearly independent elements of R_{m} that do not lie in S_{m}. It follows that for m
sufficiently large this subspace S_{m} has co-dimension exactly m in R_{m}. We thus see that the
dimension of S_{m} is m(d - 1) + 1 - g. It follows that the arithmetic genus is not changed by
projection.

Consider a curve C ⊂ ℙ^{n} of degree d; it is natural to assume that C “spans” ℙ^{n} so that it does not
lie in a linear subspace of ℙ^{n}. Such a curve is said to be non-degenerate curve in ℙ^{n} and we
will henceforth restrict our attention to non-degenerate curves. A linear equation L defines a
sub-variety of C consisting of at most d distinct points. Given any n points in ℙ^{n}, they lie in a ℙ^{n-1}.
Hence, we must have d ≥ n for a non-degenerate curve. By applying this argument to the m-th
Veronese embedding of the curve for sufficiently large m we see that md ≥ md - g or equivalently
g ≥ 0.

Upon replacing the given embedding of C by a suitable Veronese twist we can assume that the dimension
of R_{m} is dm + 1 - g for all m; note that d has been increased while ensuring this. Now, for any k ≤ d - g
and any k points on the curve, consider all hyperplanes that contain these k points. This gives a linear
system of hyperplanes of dimension at least d - k - g (it could be bigger if the k points are linearly
dependent); moreover, the residual intersection (after leaving out the fixed k point locus) is of degree d-k.
We generalise this statement below.

A divisor on a curve C is a finite formal expression of the form ∑
_{i}n_{i}P_{i} where P_{i}’s are points of C
and n_{i}’s are integers; in other words, the divisor group Div(C) is the free abelian group on
points of C. A divisor is called effective if the n_{i}’s are positive. A divisor A is said to contain a
divisor B if the difference A - B is effective. The sum ∑
_{i}n_{i} is called the degree of the divisor
∑
_{i}n_{i}P_{i}.

In an obvious way, each element F of the homogeneous co-ordinate ring R of C defines a divisor
div(F). Since every point of C is an algebraic sub-variety, it is clear that every effective divisor is
contained in a divisor of the type div(F) for some F in R. A divisor D is said to be linearly
equivalent to zero if it is of the form div(F) - div(G) where F and G lie in some R_{m} (for the same
m). Two divisors are said to be linearly equivalent if their difference is linearly equivalent to
zero. In particular, it is clear that two divisors in a linear system as defined above are linearly
equivalent.

Fix a divisor D = A-B where A and B are effective. Further, replace C by its Veronese embedding so
that A is contained in div(L) for some linear L (i. e. L lies in R_{1}), we can then write D as div(L) - B′
where B′ = B + div(L) - A is an effective divisor. The linear system of hyperplane divisors div(M) that
contain B′ is gives a collection of effective divisors that are linearly equivalent to D; moreover, as seen above
it has dimension at least deg(L) - deg(B′) - g. Thus we have proved Riemann’s inequality that the linear
system of effective divisors linearly equivalent to a divisor D is of dimension at least deg(D) - g. In
particular, given a divisor D of degree at least g there is an effective divisor that is linearly equivalent to
it.

Now, if we fix an integer k, then the k-fold Veronese embedding of ℙ^{n} → ℙ^{m} (where m = - 1) has
the property that the image of any k points of ℙ^{n} in ℙ^{m} gives a linearly independent set. (Hint: Use
Vandermonde determinant).

Let us replace C by a Veronese embedding chosen so that dim(R_{m}) = md + 1 - g and so that any g
points on C are linearly independent. Let D be any effective divisor of degree d - g. Since
dim(R_{1}) = d + 1 - g, we see that D is contained in a hyperplane divisor div(L) for some L. Moreover,
div(L) - D is a linearly independent set and so we see that the linear system of all divisors linearly
equivalent to D is of dimension precisely deg(D) -g; such divisors could be called “general” divisors since
we are interested in the special divisors for which such an inequality is strict. What we have just shown is
that there is an upper bound on the degree of all special divisors. Let us note that if D is an effective
divisor of degree less than g then it is automatically special; “some things become special just by
existing!”

The Riemann-Roch theorem for curves implies the assertion that every effective special divisor is
contained in a divisor linearly equivalent to a divisor K_{C} whose degree is 2g - 2; this divisor (or strictly
speaking its linear equivalence class) is called the canonical divisor of C. When g ≥ 2, there is a
morphism C → ℙ^{g-1} such that the inverse image of any hyperplane is a divisor linearly equivalent
to K_{C}. Special divisors on C can then be identified with effective divisors whose image lies
in a hyperplane in C. In particular, any effective divisor D of degree at most g - 1 is clearly
special. Given such a divisor D let ⟨D⟩ denote the linear span of D. In general, we will have
dim⟨D⟩ = deg(D) - 1 but in some cases ⟨D⟩ will be smaller dimensional; in other words it will be even more
special. The Riemann-Roch theorem for such divisors is the very elegant assertion that the
dimension of the linear system |D| of effective divisors linearly equivalent to D is of dimension
deg(D) - dim⟨D⟩- 1.

An open problem in the study of smooth projective curves is Green’s conjecture; this attracted the attention of S. Ramanan (a partial answer and new approach formed the basis of my Ph. D. thesis with him). He and V. Srinivas also gave a new approach to the problem. C. Voisin proved the conjecture for “most” curves a few years ago but since the conjecture deals with special divisors (and so special curves), the conjecture should still attract study from algebraic geometers. A concise statement of the conjecture is that it gives a precise relation between an explicit resolution of the co-ordinate ring of the canonical embedding of a curve and the special divisors (or as seen above linear dependent collections of points) on it.