Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 152 / 2 / a / iv

Table of contents
- Solution iv

Solution

The Cox ring has one variable for each ray and is

C [x, y, z]

, graded by

de g x = 1, de g y = a, de g z = b .

The irrelevant locus is the common zero of all three variables. The quasitorus associated with the class group is

C^{*}

and acts by

t \cdot (x, y, z) = (t x, t^{a} y, t^{b} z) .

The Cox construction therefore identifies the closed points of

P (1, a, b)

with

(A^{3} ∖ {0}) / C^{*} .

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