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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 113

For the <closed immersion> $i:Y\hookrightarrow X$, the <ideal sheaf of a closed subscheme> gives
$$
0\longrightarrow\mathcal I_{Y/X}\longrightarrow\mathcal O_X
\longrightarrow i_*\mathcal O_Y\longrightarrow0.
$$
Because the <projective hypersurface> is cut out by one <homogeneous polynomial> $f$ of degree $d$, multiplication by $f$ identifies
$$
\boxed{\mathcal I_{Y/X}\cong\mathcal O_X(-d)=\mathcal O_X(1)^{\otimes(-d)}.}
$$
This is the <ideal-sheaf sequence of a projective hypersurface>.

Solution

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

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

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

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