Multiplicity of a plane curve at a point

Let $C=V(F)\subseteq {\mathbf{P}}^2$ be a plane projective curve over the algebraically closed field $k$ Plane projective curves and their components and let $p\in {\mathbf{P}}^2$. Let $D_{+}(x_i)$ be a standard chart containing $p$, identified with ${\mathbf{A}}^2$ by the ratio coordinates, and let $f$ be the dehomogenisation of $F$ in that chart projective hypersurface affine pieces, standard projective opens are affine spaces. Choosing affine coordinates $(u,v)$ of ${\mathbf{A}}^2$ centred at $p$, expand the polynomial $f$ by total degree,

$f=f_m+f_{m+1}+\cdots +f_d,f_j\text{ homogeneous of degree }j,f_m\mathrm{\ne }0.$

The multiplicity of $C$ at $p$ is

$m_p(C):=m,$

the least degree of a term in such a centred expansion; we call $f_m$ a lowest-degree part of a local equation of $C$ at $p$. For $p\mathrm{\notin }C$ set $m_p(C)=0$, and for $p\in C$ the number $m=m_p(C)\ge 1$ is also characterised as

$m_p(C)=\max \{n\ge 0:f\in {\mathfrak{m}}_p^n\},$

where ${\mathfrak{m}}_p\subseteq {\mathcal{O}}_{{\mathbf{P}}^2,p}$ is the maximal ideal of the local ring of the plane at $p$ and $f$ is any local equation of $C$ at $p$ Germs of regular functions and the local ring at a point of a classical affine variety, The local ring at a point of an affine variety is the localization at its maximal ideal, Localisation at a prime ideal: $R_{\mathfrak{p}}=(R\setminus \mathfrak{p}{)}^{-1}R$, $R_{\mathfrak{p}}$ is local with unique maximal ideal $\mathfrak{p}{R}_{\mathfrak{p}}$.

• Well-definedness. Assume the Axiom of Choice for the cited classical-variety and defining-equation identifications The Axiom of Choice. The naming formula for the order of a fixed local equation makes no choice. The number $m_p(C)$ does not depend on the chart, on the centred affine coordinates, or on the choice of the defining form $F$. The local ring ${\mathcal{O}}_{{\mathbf{P}}^2,p}$ is intrinsic to the point Germs of regular functions and the local ring at a point of a classical affine variety; two charts containing $p$ give canonically isomorphic local rings and the two dehomogenisations of $F$ differ in ${\mathcal{O}}_{{\mathbf{P}}^2,p}$ by a unit, as do two defining forms of $C$, since $V(\lambda F)=V(F)$ and $F$ is square-free up to a scalar Plane projective curves and their components, standard projective opens are affine spaces. Multiplication by a unit preserves $\max \{n:f\in {\mathfrak{m}}_p^n\}$, and an affine change of coordinates centred at $p$ induces an automorphism of the local ring preserving its maximal ideal Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms, The local ring at a point of an affine variety is the localization at its maximal ideal. In centred coordinates write $O=k[u,v{]}_{(u,v)}$. The polynomial ideal $(u,v{)}^n$ consists exactly of polynomials with no terms of degree $<n$. If $f\in (u,v{)}^{n}O$, clearing denominators gives $sf\in (u,v{)}^n$ for some $s$ with $s(0)\mathrm{\ne }0$; its lowest nonzero homogeneous part is $s(0)f_m$, so $m\ge n$. Conversely $m\ge n$ implies $f\in (u,v{)}^n$. Thus $f\in {\mathfrak{m}}_p^n$ exactly when $n\le m$, and the maximum such $n$ is $m$; this is the same number for every centred expansion, so the lowest-degree part $f_m$ is well defined up to the choice of coordinates and generates the same line of leading forms. A unit has order $0$, a local equation of a curve through $p$ has order $\ge 1$, and $f\in {\mathfrak{m}}_p$ exactly when $f(p)=0$ Evaluation and roots of a polynomial in a commutative target ring; hence $m_p(C)\ge 1$ if and only if $p\in C$, and in particular $m_p(C)=0$ exactly for $p\mathrm{\notin }C$.
• Degree bound. The centred expansion of a dehomogenised form of degree $d=\mathrm{deg}C$ has no terms beyond degree $d$, so $1\le m_p(C)\le d$ for $p\in C$ Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$.
• Multiplicativity for a union of curves. Let $F_1,F_2$ be nonconstant square-free forms with no common factor, so that $F_1{F}_2$ is square-free and $V(F_1{F}_2)$ is again a plane projective curve with components those of $V(F_1)$ and $V(F_2)$ Plane projective curves and their components. If $f_1,f_2$ are local equations at $p$ with lowest-degree parts $f_{1,m_1},f_{2,m_2}$, then $f_1{f}_2$ is a local equation of $V(F_1{F}_2)$ and its lowest-degree part is the product $f_{1,m_1}{f}_{2,m_2}$, which is nonzero because a polynomial ring over a field is a domain Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes. Hence, when both curves pass through $p$, $m_p(V(F_1{F}_2))=m_p(V(F_1))+m_p(V(F_2))\mathrm{.}$ The hypothesis that $F_1,F_2$ have no common factor is necessary for this reading: if $F_1=F_2=x_0$ then $F_1{F}_2=x_0^2$ is not square-free, $V(F_1{F}_2)$ is the line $V(x_0)$ with $m_p=1$ at $p=[0:1:1]$, while the right-hand side is $2$. For the same reason the convention of the page forbids nonreduced defining forms, and the product formula is stated here only for unions of distinct square-free curves.