If $P_X(t)=\chi(X,\mathcal O_X(t))$ is the <Hilbert polynomial> of a pure $k$-dimensional closed subscheme, then
$$
P_X(t)=\frac{\deg X}{k!}t^k+O(t^{k-1}).
$$
Equivalently, if $H=c_1(\mathcal O(1))$, the <degree of a projective scheme> is
$$
\deg(X\subseteq\mathbb P^m)=\int_{\mathbb P^m}H^k\cap[X]=\int_XH^k.
$$
A generic complementary linear subspace meets $X$ in a nonempty zero-dimensional scheme whose length is this number. It is therefore a positive integer.

Solution

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

{wiki=Degree_of_an_algebraic_variety}

For a pure $k$-dimensional projective scheme $X\subseteq\mathbb P^m$, its degree is $k!$ times the leading coefficient of its Hilbert polynomial, equivalently $\int_XH^k$.


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