Lagmental Vicfred

Bézout Counts Plane Intersections with Multiplicity by Vicfred

Last updated: Mon 16 February 2026

tags: mathematics, projective geometry, Proj, Hilbert polynomials

Two projective plane curves without a common component have total intersection number equal to the product of their degrees. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Statement

A graded ring \(S=\bigoplus_{d\ge0}S_d\) produces \(\operatorname{Proj}S\), whose points are homogeneous primes avoiding the irrelevant ideal \(S_+\). Projective space is the basic example.

$$ C=V(F),\quad D=V(G)\subset\mathbf P_k^2 $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \sum_{p\in C\cap D}I_p(C,D)=\deg F\deg G $$

Worked algebra

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \begin{cases}\text{line}\cap\text{conic}:&1\cdot2=2,\\\text{two cubics}:&3\cdot3=9,\end{cases}\qquad\text{counted over }\overline k\text{ with multiplicity} $$

Conceptual compression

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad C=V(F),\quad D=V(G)\subset\mathbf P_k^2,\\[5pt] \mathsf{C}\;&:\quad \sum_{p\in C\cap D}I_p(C,D)=\deg F\deg G. \end{aligned} $$

Caveat

Homogeneity is essential: a nonhomogeneous equation is not well defined under rescaling of projective coordinates.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \sum_{p\in C\cap D}I_p(C,D)=\deg F\deg G \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Mon 08 September 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.