Tangent cone and tangent lines at a point

Let $C=V(F)$ be a plane projective curve over the algebraically closed field $k$, let $p\in C$, and let $m=m_p(C)\ge 1$ Plane projective curves and their components, Multiplicity of a plane curve at a point. Choose a standard chart containing $p$ and affine coordinates $(u,v)$ centred at $p$, and let $f=f_m+f_{m+1}+\cdots +f_d$ be the centred expansion, with $f_m\mathrm{\ne }0$ homogeneous of degree $m$ Multiplicity of a plane curve at a point.

The tangent cone of $C$ at $p$ is the cone $V(f_m)$ inside the tangent plane at $p$, where the tangent plane is identified with $k^2$ through $(u,v)$ and the cone is the zero set of the lowest-degree form $f_m$ The intrinsic Zariski tangent space, homogeneous polynomial and homogeneous ideal.

A tangent line of $C$ at $p$ is a line $L$ through $p$ whose defining linear form $\mathrm{\ell }$ divides $f_m$ in $k[u,v]$. Since $k[u,v]$ is a unique factorisation domain in which the irreducible elements are prime, $f_m$ has a factorisation

$f_m=c{\prod }_L{\mathrm{\ell }}_L^{r_L},c\in k^{\times },$

into pairwise nonproportional linear forms ${\mathrm{\ell }}_L$, and the multiplicity of the tangent line $L$ is the exponent $r_L\ge 1$; the factorization is unique up to the order of the factors and the choice of the scalars ${\mathrm{\ell }}_L$. In particular

${\sum }_L{r}_L=m_p(C):$

the tangent lines of $C$ at $p$, counted with multiplicity, number $m_p(C)$. When $m_p(C)=1$ there is exactly one tangent line, of multiplicity one.

• Existence of the factorization. Every nonzero binary form $f$ of degree $n\ge 1$ over the algebraically closed field $k$ is a product of linear forms: if $u\mid f$ write $f=u\cdot f^{\mathrm{\prime }}$ and use induction on $n$; otherwise $f(1,T)\in k[T]$ is a polynomial of degree exactly $n$, which by the factor theorem and the root property of algebraically closed fields is $c{\prod }_j(T-{\lambda }_j)$ for $c\in k^{\times }$ and ${\lambda }_j\in k$, whence $f(u,v)=c{\prod }_j(v-{\lambda }_{j}u)$ after comparing the two homogeneous polynomials of degree $n$ on the line $u=1$ Factor theorem over a commutative ring, Evaluation and roots of a polynomial in a commutative target ring, An algebraically closed field: every nonconstant polynomial has a root in the field. Distinct tangent lines correspond to distinct roots of $f(1,T)$ up to the factor $u^s$ removed, so the linear forms are pairwise nonproportional. Unique factorisation determines the geometric lines and their exponents $r_L$ up to order; $c$ is determined only after the representatives ${\mathrm{\ell }}_L$ are fixed, and replacing ${\mathrm{\ell }}_L$ by $a_L{\mathrm{\ell }}_L$ replaces $c$ by $c{\prod }_L{a}_L^{-r_L}$ Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Irreducible and prime elements of an integral domain, Field.
• The count. Multiplying the factor multiplicities gives $\mathrm{deg}{f}_m={\sum }_L{r}_L$ because the product of homogeneous forms of degrees $a,b$ is nonzero and homogeneous of degree $a+b$ in the polynomial domain, and $\mathrm{deg}{f}_m=m$ by definition of the multiplicity, so the tangent lines counted with multiplicity number $m_p(C)$ homogeneous polynomial and homogeneous ideal, Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes. If $m=1$ then $f_1$ is a nonzero linear form, which has exactly one linear factor up to a unit, so there is exactly one tangent line and its multiplicity is one.
• Independence of choices. A change of centred affine coordinates at $p$ acts on $(u,v)$ by an invertible linear substitution, under which a binary form of degree $m$ transforms to another binary form of degree $m$ with the same factorisation transported by the substitution; hence the set of tangent lines through $p$ and their multiplicities is unchanged as a geometric datum in the tangent plane. For a chart transition fixing $p$, write its local coordinates as an invertible linear first-order part plus terms of order at least two; the inverse transition shows that linear part is invertible. Substituting into a local equation of order $m$ changes its lowest-degree form by that linear substitution. Multiplying the local equation by a unit multiplies the initial form only by the nonzero residue of that unit. Thus chart changes transport the tangent factors by the derivative, and rescaling $F$ multiplies them by a nonzero scalar, preserving the geometric lines and exponents. The tangent cone and the tangent line multiplicities are therefore invariants of the pair $(C,p)$.