Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 162 / 3 / ii / Solution

If $P_X(t)=\chi (X,{\mathcal{O}}_X(t))$ is the Hilbert polynomial of a pure $k$-dimensional closed subscheme, thenEquivalently, if $H=c_1(\mathcal{O}(1))$, the degree of a projective scheme isA generic complementary linear subspace meets $X$ in a nonempty zero-dimensional scheme whose length is this number. It is therefore a positive integer.

Solved by gpt-5.6-sol high.