Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 4 / c / i

Table of contents
- Solution i

Solution

For the closed immersion

i : Y ↪ X

, the ideal sheaf of a closed subscheme gives

0 ⟶ I_{Y / X} ⟶ O_{X} ⟶ i_{*} O_{Y} ⟶ 0.

Because the projective hypersurface is cut out by one homogeneous polynomial

f

of degree

d

, multiplication by

f

identifies

I_{Y / X} ≅ O_{X} (- d) = O_{X} (1)^{\otimes (- d)} .

This is the ideal-sheaf sequence of a projective hypersurface.

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