Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 3 / d

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- Solution d

Solution

Choose a very ample line bundle

A

on

X

. By the theorem that a high ample twist is very ample, for sufficiently large

m

both

H_{1} = A^{\otimes m} \otimes L, H_{2} = A^{\otimes m}

are very ample. The dual bundle cancels the power of

A

L ≅ H_{1} \otimes H_{2}^{*} .

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