Local intersection multiplicity of two plane curves

Let $C=V(F)$ and $D=V(G)$ be plane projective curves over the algebraically closed field $k$ Plane projective curves and their components and let $p\in {\mathbf{P}}^2$. Assume $C$ and $D$ have no common local component at $p$: in local equations $f,g\in {\mathcal{O}}_{{\mathbf{P}}^2,p}$ of $C$ and $D$ at $p$, no irreducible element of the local ring divides both $f$ and $g$ Germs of regular functions and the local ring at a point of a classical affine variety, A local ring is a nonzero commutative ring with a unique maximal ideal, The local ring at a point of an affine variety is the localization at its maximal ideal. Then the local intersection multiplicity of $C$ and $D$ at $p$ is

$I_p(C,D):={\mathrm{\ell }}_{{\mathcal{O}}_{{\mathbf{P}}^2,p}}({\mathcal{O}}_{{\mathbf{P}}^2,p}\mathrm{/}(f,g)),$

the length of the quotient of the local ring of the plane at $p$ by the ideal generated by local equations of the two curves Composition series and length of a module. Under the Axiom of Choice, the quotient has finite length by the preceding lemma Finite local length exactly when no common local branch, so $I_p(C,D)$ is a natural number. One writes $I_p(C,D)=\mathrm{\infty }$ when $C$ and $D$ share a local branch through $p$, and $I_p(C,D)=0$ when $p\mathrm{\notin }C\cap D$. The value depends only on $C,D,p$, not on the chart or the equations chosen to compute it; this is the content of the invariance lemma Invariance of the local intersection multiplicity ↗.

• Why the hypothesis is exactly local. The condition is checked on the germs at $p$: it holds at $p$ exactly when no branch of $C$ through $p$ coincides with a branch of $D$ through $p$. Failure at $p$ means an irreducible $h\in {\mathcal{O}}_{{\mathbf{P}}^2,p}$ divides both local equations; by the local-UFD argument in Finite local length exactly when no common local branch (Proof 1.3), such an $h$ is associate to an irreducible polynomial in the affine chart vanishing at $p$. Its homogenisation $H$ is irreducible and not divisible by the chart coordinate: any homogeneous factorisation would dehomogenise to a nontrivial factorisation of $h$, unless a factor were a power of that coordinate. Divisibility of both dehomogenised equations then makes $H$ divide both $F,G$ after homogenisation (extra chart-coordinate factors cannot absorb $H$). Thus the two curves share an irreducible component $V(H)$ through $p$, and conversely a shared component through $p$ fails the condition there. Thus the hypothesis is the pointwise, local analogue of "no common component", and it is imposed only at the point $p$ under consideration. The transverse case and the general local computations are proved later on this page.
• Relation to multiplicities. Since $f\in {\mathfrak{m}}_p^m$ and $g\in {\mathfrak{m}}_p^n$ for $m=m_p(C)$, $n=m_p(D)$ Multiplicity of a plane curve at a point, the ideal $(f,g)$ lies in ${\mathfrak{m}}_p^{\min (m,n)}$, and the quotient is nonzero exactly when $p\in C\cap D$; for $p\mathrm{\notin }C\cap D$ at least one of the local equations is a unit, so the quotient is the zero ring of length $0$, matching the convention above.
• Well-definedness. Changing the local equations of $C$ and $D$ changes the ideal $(f,g)$ only by units of the local ring or by replacing the pair with another generating pair of the same ideal; changing the chart or the projective coordinates induces an isomorphism of the local ring at the corresponding point. Length is invariant under ring isomorphisms and depends only on the ideal, so the definition is independent of all choices; the lemma declared as the justification records this in full. The length formula itself is a naming convention; the asserted finite length and its supplier assume the Axiom of Choice The Axiom of Choice.