Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 126 / 3 / i

Solution

Let

m_{I} : X^{3} \to X

add the coordinates indexed by a nonempty subset

I

of the three-element index set. The Theorem of the Cube says that for every line bundle

L

m_{123}^{*} L \otimes m_{12}^{*} L^{\lor} \otimes m_{13}^{*} L^{\lor} \otimes m_{23}^{*} L^{\lor} \otimes m_{1}^{*} L \otimes m_{2}^{*} L \otimes m_{3}^{*} L

is trivial, up to the harmless constant line given by the fiber of

L

at the identity.

Pull this line bundle back along

(f, g, h) : Y \to X^{3}

. Pullback commutes with tensor products and duals, and

m_{I} \circ (f, g, h)

is the corresponding sum of morphisms. The resulting bundle is precisely

M_{f, g, h}

, so it is trivial.

Take

Y = X

f = id_{X}

, and let

g, h

be the constant maps with values

x, y

. All pullbacks along constant maps are trivial line bundles. The formula for

M_{f, g, h}

then becomes

T_{x + y}^{*} L \otimes (T_{x}^{*} L)^{\lor} \otimes (T_{y}^{*} L)^{\lor} \otimes L ≃ O_{X},

or equivalently

T_{x + y}^{*} L ≃ T_{x}^{*} L \otimes T_{y}^{*} L \otimes L^{\lor} .

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