Now, the way that the Riemann-Roch theorem was phrased before, the geometry wasn’t obvious. We had to extract it in terms of rational functions with poles given by a divisor. Now that we’ve talked about canonical curves, we can use them to give a more obviously geometric form of the Riemann-Roch Theorem.

To get to the geometric statement, we start by taking an effective divisor D=p_1+\ldots+p_d with the p_i not necessarily distinct. If \phi:C\to \mathbb{P}^r is a map of the curve into projective space, we define \overline{\phi(D)} to be the intersection of all the hyperplanes which contain \phi(D). What that means is that the divisor on the curve given by \sum_{p\in C} \mathrm{mult}_p(H\cap C) p=\phi^*(H) has \phi^*(H)-D an effective divisor. We also say that \phi^*(H) contains D.

What this really means is that if all the points are distinct, then we just take the intersection of all hyperplanes containing them all. But if some have multiplicities, we require that the planes intersect the curve in that point with multiplicity, so if two are equal, then the hyperplanes must be at least simply tangent to the point, etc.

Now, the divisors in the linear system |K| are the same as those cut out by hyperplanes on the canonical curve. Now, for a given divisor in |K| to contain D is just a set of linear conditions, so we have a linear subsystem of |K|. Every divisor in this system contains D, so D is the base locus of this system. We can remove the base locus by looking at the system |K-D|, so every hyperplane containing D is in |K-D|. But if E\in |K-D|, then D+E\in |K|, so the elements of |K-D| are in 1-1 correspondence with the hyperplanes containing D.

The set of hyperplanes containing D is a subspace of (\mathbb{P}^{g-1})^*, and the dimension of this space will be complementary to that of \overline{\phi_K(D)}. So we have \dim |K-D|+\dim \overline{\phi_K(D)}=g-2 (the g-2 is because we want to lift them to affine g-dimensional space, in which case we get the left hand side, plus two, is equal to g). Solving this, we get \dim |K-D|=g-2-\dim\overline{\phi_K(D)}. Plugging this into the Riemann-Roch Theorem, we get

Geometric Riemann-Roch: Let C be a curve of genus g, and D an effective divisor. Then \dim |D|=\deg D-1-\dim\overline{\phi_K(D)}.

The word general gets thrown around a lot, so remember that it gets defined in specific contexts, though usually means that a statement holds for some dense open subset of the objects being considered. Here, we’ll say a divisor is general if \overline{\phi_K(D)} has maximum possible dimension. Then, for a general effective divisor with \deg D\leq g, we have h^0(D)=1 and h^1(D)=h^0(K-D)=d-g, so that |D|=\{D\}. This is because the dimension of the span of \deg(D) points will be a \deg D-1 plane, and so Geometric Riemann-Roch says \dim |D|=0.

Similarly, if \deg D\geq g, we have h^1(D)=0 and h^0(D)=d+1-g, because \dim\overline{\phi_K(D)}=g-1, so we get d-(g-1)=d-g+1.

Geometric Riemann-Roch simplifies quite a few arguments, and is used often in proving things about curves. Plus, it really brings out the moral that the geometry of the canonical curve is intrinsic.

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