Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 152 / 3 / c

Solution

The two standard affine charts of the blowup restrict to an affine open cover of

X_{0}

. On the

x

-chart write

y = xu

. Since

x y = x^{2} u

, the fiber is

U_{x} \cap X_{0} = Spec k [x, u] / (x^{2} u) .

On the

y

-chart write

x = y v

U_{y} \cap X_{0} = Spec k [y, v] / (y^{2} v) .

On their overlap,

u

and

v

are invertible with

v = u^{- 1}

and

y = xu

. The squared factors

x^{2}

and

y^{2}

display the doubled exceptional component explicitly.

Solved by gpt-5.6-sol high.

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