Multiplicity one characterises smooth points with a unique tangent

In the chart, $f$ is square-free with $f(p)=0$, and the expansion around $p$ is $f=f_m+f_{m+1}+\cdots$ with $m=m_p(C)\ge 1$ and $f_m\mathrm{\ne }0$. The partial derivatives satisfy $f_u,f_v\in {\mathfrak{m}}_p^{m-1}$: differentiating a degree-$j$ term lowers its order by one. Hence if $m\ge 2$ both partial derivatives vanish at $p$, while if $m=1$ the lowest part $f_1=au+bv$ is a nonzero linear form and $(f_u(p),f_v(p))=(a,b)\mathrm{\ne }(0,0)$.

The lowest part $f_m$ is a line if and only if $m=1$, and $V(f_m)$ is a single line of multiplicity one exactly when the factorisation of [F3] has one factor with exponent one, which happens exactly when ${\sum }_L{r}_L=m=1$. Thus (1) and (3) are equivalent: a single tangent line of multiplicity one is a single nonzero linear form, and conversely if $m\ge 2$ the sum of the multiplicities is at least two, giving either at least two tangent lines or one line of multiplicity at least two.

The Jacobian matrix of the single equation $f$ at $p$ is the $1\times 2$ matrix $(f_u(p),f_v(p))$, whose rank over $k$ is $0$ or $1$. By [F2] and [F5], $A_{\mathfrak{n}}=O\mathrm{/}(f)$ has dimension $1$, so the rank is $2-\dim A_{\mathfrak{n}}=1$ if and only if $A_{\mathfrak{n}}$ is regular. By step 1.1 the rank is $1$ if and only if $m=1$. Therefore (1) if and only if $p$ is regular, and regularity is independent of the chart and the defining form because $m_p(C)$ is Multiplicity of a plane curve at a point. To compare with the homogeneous derivatives, take a representative with $x_i=1$. The affine derivatives are the other two homogeneous partials evaluated there; the termwise Euler identity ${\sum }_j{x}_j{F}_{x_j}=dF$ gives $F_{x_i}(p)=-{\sum }_{j\mathrm{\ne }i}{x}_j(p)F_{x_j}(p)$ because $F(p)=0$. Thus all three homogeneous partials vanish exactly when both affine ones do, in every characteristic, without dividing by $d$. This proves (2) and identifies the singular points of the chart as those with $m\ge 2$.

Assume $m=1$. By step 1.2 and [F3] there is exactly one tangent line and its multiplicity is one; by step 2.1 the local ring ${\mathcal{O}}_{C,p}$ is a one-dimensional regular local ring. If $m\ge 2$, step 2.1 shows $p$ is singular. In the chart the singular points are the common zeros of $f,f_u,f_v$, a closed subset of the affine curve. It is proper: not all partial derivatives of the square-free $f$ can vanish, because in characteristic zero that would force $f$ to be constant, while in characteristic $p>0$ both vanishing partials would make $f=g^p$ a $p$-th power by [F7], and a nonconstant $p$-th power is not square-free Irreducible and prime elements of an integral domain; so after interchanging $u,v$ if necessary, some partial $f_u\mathrm{\ne }0$, its degree is at most $\mathrm{deg}f-1<\mathrm{deg}f$, hence $f\nmid f_u$ and by [F6] $f_u$ does not vanish on all of $V(f)$. Covering $C$ by the three standard charts and using the chart-independence of the multiplicity, the singular locus of $C$ is closed in $C$ and not all of $C$.