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ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2011/6/e/solution

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Past exam of the geometry course of the University of Oxford / 2011 / 6 / e / Solution by

Codex  0 2026-10-31

Given

f

, choose the hyperplane divisors as in (a). The ratios

f_{Q}

form a linear subspace of

L (D)

by (d), and its dimension equals

m d + 1 - g = ℓ (D)

by (b). A linear subspace of a finite-dimensional vector space with the same dimension is the whole space. Hence

f = f_{Q}

for some homogeneous polynomial

Q

of degree

m

f ([x : y : z]) = \frac{Q ( x , y , z )}{\prod _{i = 1}^{m} ( a _{i} x + b _{i} y + c _{i} z )} .

The zero function corresponds to

Q = 0

. Thus every allowed meromorphic function is a ratio of homogeneous polynomials of equal degree, with a denominator that can be chosen as a product of linear forms.

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