Plane Curves, Local Intersection Multiplicity, and Bézout

Work over an algebraically closed field $k$ and fix $d\ge 1$. The space $k[x_0,x_1,x_2{]}_d$ of homogeneous degree-$d$ forms has as a basis the monomials $x_0^i{x}_1^j{x}_2^{d-i-j}$ with $i,j\ge 0$, $i+j\le d$ homogeneous polynomial and homogeneous ideal, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$. The bijection $(i,j)\mapsto \{i,i+j+1\}\subseteq \{0,\dots ,d+1\}$ shows that its dimension is $\left(\genfrac{}{}{0px}{}{d+2}{2}\right)$ The set $[A{]}^k$ of $k$-element subsets and the binomial coefficient $\left(\genfrac{}{}{0px}{}{n}{k}\right):=\mid [n{]}^k\mid$, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis.

A linear system of plane curves of degree $d$ is a linear subspace $W\subseteq k[x_0,x_1,x_2{]}_d$ Linear subspace of a vector space, Vector space over a field. Its elements are the nonzero forms modulo nonzero scalar multiplication, each equipped with its projective zero set:

$[F]⟼V_{+}((F))\subseteq {\mathbf{P}}_k^2,0\mathrm{\ne }F\in W\mathrm{.}$

Here $[F]$ denotes its scalar-equivalence class; scaling does not change its zero set projective space points, projective algebraic set. The dimension of the system is ${\dim }_{k}W-1$. The zero subspace is allowed as the empty system, assigned dimension $-1$ by convention. A pencil has dimension one, equivalently ${\dim }_{k}W=2$.

The base locus is

$\mathrm{Bs}W={\bigcap }_{0\mathrm{\ne }F\in W}V(F)\mathrm{.}$

It is closed as an intersection of projective closed sets projective zariski topology; the empty system has base locus all of ${\mathbf{P}}_k^2$. For a finite set $S\subseteq {\mathbf{P}}_k^2$, the subsystem through $S$ is

$W(-S)=\{F\in W:F(q)=0\text{ for every }q\in S\}\mathrm{.}$

The condition is independent of the chosen nonzero representatives of $q$, since $F(\lambda q)={\lambda }^{d}F(q)$, and is linear in $F$, so $W(-S)$ is a subspace. For the full space $W=k[x_0,x_1,x_2{]}_d$, the dimension is $\left(\genfrac{}{}{0px}{}{d+2}{2}\right)-1$ and the base locus is empty: at any projective point some $x_i\mathrm{\ne }0$, so $x_i^d$ does not vanish there. For $d=1$, a subsystem through three noncollinear points is empty, since the zero set of any nonzero linear form is a line and cannot contain all three.

• This definition retains multiplicities in degree-$d$ equations. A square-free member has reduced degree $d$ in the plane-curve convention degree projective hypersurface. A nonsquarefree form defines the same zero set as its square-free part, whose reduced degree may be smaller. For example $W=⟨x_0^2⟩$ is a degree-two equation system supported on a reduced line of degree one.
• All constructions use explicit finite-dimensional linear algebra and scalar equivalence; no choice principle is used.

Fix an algebraically closed field $k$ An algebraically closed field: every nonconstant polynomial has a root in the field and ${\mathbf{P}}^2={\mathbf{P}}_k^2$ projective space points. A (plane) projective curve is a closed subset $C=V(F)\subseteq {\mathbf{P}}^2$ given by a nonconstant square-free homogeneous form $F\in k[x_0,x_1,x_2]$ homogeneous polynomial and homogeneous ideal, projective algebraic set, projective zariski topology. Its degree is the total degree $\mathrm{deg}F$ Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$, and its irreducible components are the closed sets $V(h)$ for the distinct irreducible factors $h$ of $F$, each taken once. Thus every component is reduced and occurs with multiplicity one, $C$ may be reducible, and $C$ is nonempty. A closed set $V(F)$ with $F$ nonconstant and square-free is written $V(F)$, and we call $F$ a defining form of $C$; under the Axiom of Choice, it is determined by $C$ up to a nonzero scalar: the square-free principal ideal $(F)$ is radical, and the homogeneous radical-ideal correspondence recovers it from $V(F)$ projective irreducibility homogeneous prime, The Axiom of Choice; equality of principal ideals in the polynomial domain makes their generators associates, and its units are the nonzero constants.

An affine plane curve is $V(f)\subseteq {\mathbf{A}}^2$ for a nonconstant square-free $f\in k[x,y]$ An affine algebraic set in affine space, with its components defined by the irreducible factors of $f$ in the same way. On a standard chart $D_{+}(x_i)\cong {\mathbf{A}}^2$ the trace of a projective curve is the zero set of its dehomogenisation, which is an affine curve when the dehomogenisation is nonconstant and is empty when it is a nonzero constant: $C\cap D_{+}(x_i)=V(f_i)$ for $f_i=F(x_i\mapsto 1,x_j\mapsto x_j\mathrm{/}{x}_i\text{ for }j\mathrm{\ne }i)$ projective hypersurface affine pieces, standard projective opens are affine spaces.

Square-freeness and components. The polynomial ring $k[x_0,x_1,x_2]$ is a unique factorisation domain Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, so $F$ has a factorisation into pairwise nonassociate irreducibles, and each factor of a homogeneous form is homogeneous: the lowest and highest nonzero graded parts of a product are the products of the corresponding parts, nonzero in this domain, so a product supported in one degree forces each factor to be supported in one degree Nonnegatively graded rings and modules, homogeneous elements, and twists. Square-freeness of $F$ says that no irreducible factor is repeated: $F=c{\prod }_j{h}_j$ with $c\in k^{\times }$ and the $h_j$ pairwise nonassociate irreducible homogeneous forms. The irreducible components of $C=V(F)$ are the curves $V(h_j)$, one for each distinct factor: the union of the $V(h_j)$ is $C$, each $V(h_j)$ is irreducible and for irreducible homogeneous $h$, the homogeneous vanishing ideal of $V(h)$ is $(h)$ by the radical-ideal correspondence, so up to units and order the list is determined by $F$ and its members are exactly the maximal irreducible closed subsets of $C$ Irreducible topological spaces and irreducible subsets in the subspace topology, projective irreducibility homogeneous prime. Since every $h_j$ occurs exactly once, $C$ is reduced and no component carries an invisible multiplicity in its defining form.

• Nonemptiness of $C$. A nonconstant homogeneous $F$ has a nontrivial zero. First, $k$ is infinite: if $k=\{a_1,\dots ,a_q\}$ were finite, then $p(T)={\prod }_{i=1}^q(T-a_i)+1$ is nonconstant of degree $q\ge 1$ and $p(a)=1\mathrm{\ne }0$ at every $a\in k$, contradicting algebraic closure An algebraically closed field: every nonconstant polynomial has a root in the field, Evaluation and roots of a polynomial in a commutative target ring, Over an integral domain, degrees add under multiplication of nonzero polynomials. Choose a variable, say $x_2$, occurring in $F$ with exponent $r\ge 1$ and write $F={\sum }_{j=0}^r{a}_j(x_0,x_1)x_2^j$ with $a_r\mathrm{\ne }0$; the polynomial $a_r(X,1)\in k[X]$ is then nonzero, so since $k$ is infinite it is not the zero function: a nonzero polynomial of degree $n$ has at most $n$ roots, so it cannot vanish at every element of the infinite field $k$ A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots; hence there is $u\in k$ with $a_r(u,1)\mathrm{\ne }0$. The polynomial $F(u,1,T)\in k[T]$ has degree $r\ge 1$, hence has a root $c$ An algebraically closed field: every nonconstant polynomial has a root in the field, and $[u:1:c]$ is a nonzero point of $V(F)$. For a plane projective curve we use $r\ge 1$, so the argument applies; note $\mathrm{deg}F\ge 1$ throughout.
• Infinitude. Every such curve has an affine chart with a nonconstant equation $f(x,y)$. If $f$ has positive degree in $y$, its top coefficient in $k[x]$ is nonzero and vanishes at only finitely many $a\in k$; for each of the infinitely many remaining $a$, the nonconstant polynomial $f(a,y)$ has a root. If $f$ depends only on $x$, any root gives an entire affine line of zeros. Thus every plane projective curve, including each component, has infinitely many points.
• Scaling and the square-free convention. $V(\lambda F)=V(F)$ for every $\lambda \mathrm{\ne }0$, so the curve does not see the scalar, and writing $F$ without repeated factors is a genuine normalisation: with the nonreduced form $x_0^2$ the line $V(x_0)$ would otherwise be assigned the degree of a nonreduced equation. The definition therefore fixes square-free forms and records reduced components, exactly as degree projective hypersurface does for hypersurfaces.
• Where choice enters. The naming convention for a plane projective curve, the degree of a fixed defining form, and its factorisation are choice-free. The supplied radical-ideal correspondence uses the Axiom of Choice through the Nullstellensatz input of projective irreducibility homogeneous prime The Axiom of Choice. It is used both to recover the defining form up to scalar from the underlying closed set and to identify the $V(h_j)$ as its irreducible components. Consumers invoking either identification, including the resulting well-definedness of degree from the closed set, inherit AC; calculations with a fixed defining form alone need no choice.

Let $k$ be a field and let $F,G\in k[x_0,x_1,x_2]$ be nonzero homogeneous forms of positive total degrees $d,e$ homogeneous polynomial and homogeneous ideal, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$. Write $F={\sum }_{i=0}^d{a}_i(x_0,x_1)x_2^i$, $G={\sum }_{i=0}^e{b}_i(x_0,x_1)x_2^i$. Over $R=k[x_0,x_1]$ introduce auxiliary variables $X,Y$ and homogenise with the nominated degrees:

$H_F(X,Y)={\sum }_{i=0}^d{a}_i{X}^i{Y}^{d-i},H_G(X,Y)={\sum }_{i=0}^e{b}_i{X}^i{Y}^{e-i}\mathrm{.}$

Define the resultant eliminating $x_2$ by

${\mathrm{Res}}_{x_2}(F,G):={\mathrm{Res}}_{d,e}(H_F,H_G)\in k[x_0,x_1],$

using the Sylvester determinant and ordered bases of Sylvester resultant of two positive-degree binary forms. Thus $H_F(X,1)=F(x_0,x_1,X)$, but the nominations remain $d,e$ even if the actual degrees in $X$ drop.

The resultant is zero or homogeneous of total degree $de$ in $x_0,x_1$. To see this, index matrix rows by the exponent $r=0,\dots ,d+e-1$ of $X$ in the target monomial, and index the $F$-columns by multiplier exponents $j=0,\dots ,e-1$ and the $G$-columns by $j=0,\dots ,d-1$. An $F$-entry is $a_{r-j}$ of degree $d-r+j$ and a $G$-entry is $b_{r-j}$ of degree $e-r+j$. Every nonzero determinant term therefore has total degree

$ed+de+\frac{e(e-1)}{2}+\frac{d(d-1)}{2}-\frac{(d+e-1)(d+e)}{2}=de\mathrm{.}$

Specialisation. Coefficient specialisation sends the determinant to the determinant of $H_F(a,b;X,Y)$ and $H_G(a,b;X,Y)$, for any $(a,b)\mathrm{\ne }(0,0)$ Scaling, specialization, and the affine and infinite charts of a binary resultant. Over an algebraically closed extension, its vanishing detects a common projective root of these binary forms The binary Sylvester resultant detects a common geometric projective root. Such a root is either $[c:1]$, with $F(a,b,c)=G(a,b,c)=0$, or $[1:0]$, when both coefficients $a_d,b_e$ vanish. Consequently the value need not detect a finite root when both leading coefficients vanish. The later detection lemma excludes that case by assuming $[0:0:1]$ lies on neither curve.

• The elimination coordinate is fixed; the resultant polynomial depends on it. The underlying projective curves do not depend on a choice of coordinates.
• The scaling rule gives ${\mathrm{Res}}_{x_2}(uF,vG)=u^e{v}^d{\mathrm{Res}}_{x_2}(F,G)$ for $u,v\in k$ Scaling, specialization, and the affine and infinite charts of a binary resultant. In particular it is not a scalar-independent function of the curves.

Assume the Axiom of Choice. Let $k$ be a field, let $p\in {\mathbf{A}}^2(k)$, put $R=k[x,y]$, $\mathfrak{m}={\mathfrak{m}}_p$, and $O=R_{\mathfrak{m}}$, and let $f,g\in \mathfrak{m}$ be nonzero. Then the following are equivalent: (1) $(f,g)O$ is $\mathfrak{m}$-primary; (2) $O\mathrm{/}(f,g)O$ has finite length as an $O$-module; (3) $f$ and $g$ have no common irreducible factor $h\in k[x,y]$ with $h(p)=0$, equivalently no height-one prime of $R$ contained in $\mathfrak{m}$ contains $(f,g)$. If the conditions fail the length is infinite, and if they hold $(f,g)O$ is a parameter ideal of the two-dimensional regular local ring $O$, so $O\mathrm{/}(f,g)O$ is a zero-dimensional local ring of finite length. Moreover $O$ is a UFD. Every ideal $I\subseteq O$ with radical $\mathfrak{m}O$ contains a power of $\mathfrak{m}O$, and its quotient satisfies ${\mathrm{\ell }}_O(O\mathrm{/}I)={\dim }_k(O\mathrm{/}I)$. For a surjective ring map $O\to V$ and a $V$-module $M$, the submodules over $O$ and $V$ coincide, so its composition length is unchanged.

Let $k$ be a field, $R=k[x,y]$, $\mathfrak{m}=(x,y)$ and $O=R_{\mathfrak{m}}$. For every integer $t\ge 0$ the natural map $R\mathrm{/}{\mathfrak{m}}^t\to O\mathrm{/}{\mathfrak{m}}^{t}O$ is an isomorphism, and

${\mathrm{\ell }}_O(O\mathrm{/}{\mathfrak{m}}^{t}O)={\dim }_k(O\mathrm{/}{\mathfrak{m}}^{t}O)=\left(\genfrac{}{}{0px}{}{t+1}{2}\right)=\frac{t(t+1)}{2}\mathrm{.}$

The ring $O\mathrm{/}{\mathfrak{m}}^{t}O$ has a composition series whose factors are the one-dimensional $k$-vector spaces spanned by the monomials $x^i{y}^j$ with $i+j=t-1,t-2,\dots ,0$.

• The case $t=0$. Here ${\mathfrak{m}}^0=R$ and both sides of the isomorphism are the zero ring, of length $0=\left(\genfrac{}{}{0px}{}{1}{2}\right)$; the composition series is empty. The statement includes $t=0$ so that the truncation maps of the later proofs are defined without a separate convention.
• Choice. The argument is choice-free: it uses only the explicit monomial basis, the finite geometric sum, and the universal property of localisation. No maximal ideals are selected and no proper-ideal-into-maximal-ideal principle is invoked.

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.

Assume the Axiom of Choice. Let $C=V(F)$ and $D=V(G)$ be plane projective curves of degrees $d,e\ge 1$ over the algebraically closed field $k$ with no common component, and put $X=\mathrm{Proj}(k[x_0,x_1,x_2]\mathrm{/}(F,G))$ Projective scheme of a homogeneous quotient and its standard affine charts. Then the total length of $X$ is ${\mathrm{len}}_k(X)=de$, and since every point of $X$ is $k$-rational (the residue field is a finite extension of the algebraically closed field $k$) this reads

${\sum }_{x\in X}{\mathrm{\ell }}_{{\mathcal{O}}_{X,x}}\left({\mathcal{O}}_{X,x}\right)=de\mathrm{.}$

Let $F,G\in k[x_0,x_1,x_2]$ be nonzero homogeneous forms of positive degrees $d,e$ over the algebraically closed field $k$ with no common factor, and suppose the point $[0:0:1]$ lies on neither $C=V(F)$ nor $D=V(G)$; equivalently the coefficient of $x_2^d$ in $F$ and the coefficient of $x_2^e$ in $G$ are nonzero constants, so that $F(a,b,x_2)$ and $G(a,b,x_2)$ have exact degrees $d,e$ for every $(a,b)\mathrm{\ne }(0,0)$. Then ${\mathrm{Res}}_{x_2}(F,G)\in k[x_0,x_1]$ is a nonzero form of degree $de$, and for $(a,b)\mathrm{\ne }(0,0)$ one has ${\mathrm{Res}}_{x_2}(F,G)(a,b)=0$ if and only if there is $c\in k$ with $[a:b:c]\in C\cap D$. Consequently the projections $(x_0:x_1)$ of the common points of $C$ and $D$ are exactly the projective zeros of the resultant; over each such zero the fibre of common points is finite, of size at most $\min (d,e)$ when counted without multiplicity. In particular $C\cap D$ is finite.