Past exam of the geometry course of the University of Oxford / 2011 / 6 / d

Write $L_i=a_{i}x+b_{i}y+c_{i}z$ and $S={\prod }_i{L}_i$. The numerator and denominator have the same homogeneous degree, so their ratio is well-defined on projective points away from its poles and extends as a meromorphic function on $C$. Locally the pole order cannot exceed the zero order of $S{|}_C$, whose divisor is $D$. Thus $f_Q\in L(D)$.

Equality of two ratios is equivalent to $Q-Q^{\prime }$ vanishing on $C$. A nonsingular projective plane curve is irreducible: distinct positive-degree components would intersect by Bézout's theorem, producing a singular point. Its homogeneous vanishing ideal is therefore generated by its irreducible defining polynomial $P$. One can also see this using Bézout's theorem: a homogeneous polynomial not divisible by $P$ has only finitely many intersections with $C$, so cannot vanish on the whole curve. ConsequentlyMultiplication by the nonzero polynomial $P$ is injective. The linear map $V_m\to L(D)$ therefore has kernel $P{V}_{m-d}$ and image dimensionThe assumption $m\ge d$ makes both polynomial-space dimensions applicable directly.