Geometric interpretation of the Riemann-Roch for curves

Let X be a smooth projective curve of genus g2 over an algebraically closed field k and denote by K a canonical divisor.

I have some clues about the geometrical interpretation of the Riemann-Roch Theorem for smooth algebraic curves, but also some doubts which I would like to clarify. Recall that the RR formula is
h0(X,D)h0(X,KD)=dg+1.

Assume that X is not hyperelliptic, so that the canonical map is actually a canonical embedding
ϕK:XPg1P{sH0(X,K)s(P)=0}
giving a preferred realization of the curve inside a (g1)-dimensional projective space.

The key feature of such an embedding is that there is a bijective correspondence between hyperplanes WPg1 and effective divisors in the linear system |K|PH0(X,K).

The picture shows the canonical embedding in P2 of a non hyperelliptic curve of genus 3.

The canonical embedding in the plane of a non hyperelliptic curve of genus 3

Let D=di=1Pi be an effective divisor consisting of d<g distinct points of X. We define
ϕK(D):=span{ϕK(P1),,ϕK(Pd)}.

The vector space H0(X,KD) can be interpreted as the space of canonical divisors containing D, and here comes my first question:

(1) Is it correct to identify PH0(X,KD) with the set of hyperplanes of Pg1 passing through ϕK(D) ? If so, how can one see it formally?

Let r(D):=dim|D| denote the dimension of the complete linear series associated to D. Further, denote by D=KD the residual divisor of degree d=2g2d.

If (1) is correct, then it follows that r(D) equals the number of hyperplanes of Pg1 passing through ϕK(D). Now, notice that the RR can be rewritten as

r(D)=[g1][dr(D)]

so that we deduce that r(D) counts the number of independent linear relations on the points of D and we can give the following geometrical interpretation of the Riemann-Roch:

The integer r(D) is the number of hyperplanes passing though ϕK(D), hence it equals the difference between the dimension g1 of the ambient space and the dimension of the space spanned by the points of ϕK(D).

Of course my second question is:

(2) Do you agree with this geometrical interpretation?

Answer

The answer to your first question is yes. As for the second question I would make a minor adjustment but the idea is correct. Please be patient with me as I set things up first.

Notation

Let ωC be the sheaf of holomorphic differential forms on C. For any vector space V I will denote by |V| the projective space of lines in V and by P(V) the projective space of codimension 1 planes in V, henceforth referred to as hyperplanes.

Throughout I will let V=H0(C,ωC) stand for the global holomorphic differentials on C. For a divisor D on C let V(D)=H0(C,ωC(D)) which is to be viewed as a subspace of V using the natural injection ωC(D)ωC.

Let me write the map φ:CPg1 with the proper notation, because as things stand the notation suggests that generators for V have been chosen. The preferred version, as you put it, can be written as φ:CP(V) where
pVp:=ker(VωCωC|p).
It is clear that V(p)Vp but Riemann-Roch says that the dimensions match so we get V(p)=Vp.

Basic observations

Given qP(V) we get a hyperplane Hq|V| by duality. Here, Hq parametrizes hyperplanes in P(V) containing q. Given two points q1,q2 the line between q1 and q2 can be viewed as the intersection of all hyperplanes containing qi's. This is the dual of Hq1Hq2. And similarly for more points. So far this is linear algebra, let us apply it to our specific setting.

Our description of the map φ ensures that Hφ(p)=|V(p)|. Given p1,p2C the line between φ(p1) and φ(p2) has dual |V(p1)||V(p2)| which is easily seen to be |V(p1p2)| (Check!). And if D is reduced, this argument is sufficient to conclude that |V(D)||V| parametrizes the hyperplanes through φ(D).

It is possible, but difficult, to describe the locus of hyperplanes having high order contact at a point pC using just geometry. So we go back to doing a little more tautology.

Advanced tautology

If you give me any divisor D|ωC|=|V| then this gives a line in H0(C,ωC). Pick a section σ generating this line. By the standard relations D=(σ)0. Now this point D has a dual hyperplane WDP(V) consisting of all hyperplanes in |V| containing D. I claim that WDφ(C)=D.

Indeed, WD corresponds to a divisor in the linear system of the tautological line bundle O(1) which by construction has global sections canonically isomorphic to V. Furthermore, the divisor WD corresponds precisely to the (line generated by) σV. Now the intersection of WD with C can be obtained by pulling back (O(1),σ) to C. However, the pullback of O(1) is of course (canonically isomorphic to) ωC and the section σ maps to σ again by canonical identifications. This proves the desired statement.

Wrapping up

So far what we have been doing was largely exploratory. Now let's tackle your question head on. The inclusion V(D)V identifies sections of ωC(D) with sections of ωC that vanish on D (this follows from the exact sequence obtained from ωC(D)ωC). Therefore if we take σV(D) then the corresponding hyperplane HP(V) satisfies HCD. Conversely, any hyperplane that satisfies HCD corresponds to (the line generated by) a section of ωC vanishing on D, hence to a section of ωC(D). This answers your first question.

As for the second question, if d=degD and d<g then we expect φ(D) to span a d1 dimensional projective subspace. Then your calculation shows that r(D) in fact measures the failure of our expectation. More precisely, r(D)=d1dim(span(φ(D))). The space of hyperplanes in Pg1 passing through a k-dimensional projective space has dimension (g2)k. Then putting these together we get:
r(D)=(g2)(d1r(D))=(g1)(dr(D))

Attribution
Source : Link , Question Author : Abramo , Answer Author : Emre

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