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

Table of contents
- Solution ii

Solution

If

P_{X} (t) = χ (X, O_{X} (t))

is the Hilbert polynomial of a pure

k

-dimensional closed subscheme, then

P_{X} (t) = \frac{d e g X}{k !} t^{k} + O (t^{k - 1}) .

Equivalently, if

H = c_{1} (O (1))

, the degree of a projective scheme is

de g (X \subseteq P^{m}) = \int_{P^{m}} H^{k} \cap [X] = \int_{X} H^{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.

Solved by gpt-5.6-sol high.

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