Intersection multiplicity dominates the product of multiplicities, with equality for separated tangent cones

Given: AC, plane curves $C=V(F)$, $D=V(G)$ over the algebraically closed field $k$, a point $p$ with no common local component, local equations $f,g$ at $p$, and $m=m_p(C)$, $n=m_p(D)$.

In the local ring $O={\mathcal{O}}_{{\mathbf{P}}^2,p}$ with maximal ideal $\mathfrak{m}$, the orders of $f,g$ are $m,n$, and $(f,g)O$ is $\mathfrak{m}$-primary: some power ${\mathfrak{m}}^N$ lies in $(f,g)O$ Finite local length exactly when no common local branch, Multiplicity of a plane curve at a point, A local ring is a nonzero commutative ring with a unique maximal ideal.

Choose centred chart coordinates $x,y$ and the polynomial dehomogenisations $f,g$, so $O=k[x,y{]}_{(x,y)}$. Write $R=k[x,y]$ with $I=(x,y)$ and $J=I^{m+n}+(f,g)$. Then $V(J)=\{0\}$; hence $J$ is $I$-primary, $R\mathrm{/}J\cong O\mathrm{/}JO$, and ${\dim }_k(R\mathrm{/}J)={\dim }_k(O\mathrm{/}JO)$: passing to the quotient by an $I$-primary ideal makes $R\mathrm{/}J$ a local ring with maximal ideal $I\mathrm{/}J$, so its localisation at $I$ is an isomorphism The radical of an ideal, Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals, Assuming the Axiom of Choice, a local ring $R$ is canonically isomorphic to $R_{\mathfrak{m}}$ at its maximal ideal, Localisation commutes with quotient rings: $S^{-1}R\mathrm{/}{S}^{-1}I\cong {\bar{S}}^{-1}(R\mathrm{/}I)$. The finite-length local quotients here have $k$-dimension equal to their $O$-length by the finite maximal-ideal filtration in Finite local length exactly when no common local branch (Proof 1.3). Since $J\supseteq (f,g)$, the quotient $O\mathrm{/}JO$ is a quotient of $O\mathrm{/}(f,g)$, so ${\dim }_k(O\mathrm{/}JO)\le I_p(C,D)$ with equality if and only if $I^{m+n}\subseteq (f,g)O$ Local intersection multiplicity of two plane curves.

The truncated multiplication map $\bar{\psi }:R\mathrm{/}{I}^n\times R\mathrm{/}{I}^m\to R\mathrm{/}{I}^{m+n}$, $\bar{\psi }(A,B)=Af+Bg$, is well defined and $k$-linear, its image is exactly the kernel of the natural map $\varphi :R\mathrm{/}{I}^{m+n}\to R\mathrm{/}(I^{m+n},f,g)$, and $\bar{\psi }$ is injective if and only if the lowest forms $f^{\ast },g^{\ast }$ have no common factor The truncated multiplication map is injective exactly when the tangent cones are coprime. The dimension of $R\mathrm{/}{I}^t$ is $\left(\genfrac{}{}{0px}{}{t+1}{2}\right)$, and $\dim \mathrm{im}\bar{\psi }=\dim (R\mathrm{/}{I}^n)+\dim (R\mathrm{/}{I}^m)-\dim \ker \bar{\psi }$ Lengths of truncated plane local rings, Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Vector space over a field.

The tangent lines of $C$ at $p$ are the lines whose defining linear forms divide $f^{\ast }$; since $k$ is algebraically closed, $f^{\ast },g^{\ast }$ have a common factor if and only if they have a common linear factor Tangent cone and tangent lines at a point.

AC is assumed; it enters through the Nullstellensatz, primarity and localisation suppliers The Axiom of Choice, Module length is additive in short exact sequences.

If the lowest-degree forms $f^{\ast },g^{\ast }$ are coprime then $I^{m+n}\subseteq (f,g)O$, since $I^t\subseteq (f,g)O$ for every $t\ge m+n-1$ Coprime tangent cones force a power of the maximal ideal into the local ideal. The morphism $O\mathrm{/}(f,g)\to O\mathrm{/}JO$ is the quotient by the image of $I^{m+n}$, so [F2] makes the equality $I_p(C,D)={\dim }_k(R\mathrm{/}J)$ hold whenever the initial forms are coprime. The containment may also hold for other initial forms; the equality $I_p=mn$ additionally requires injectivity of the truncated syzygy map, as shown below.