Uniformising parameters at smooth points of a plane curve

Assume the Axiom of Choice. Let $C=V(F)$ be a plane projective curve over the algebraically closed field $k$ and let $p\in C$ be a smooth point Multiplicity one characterises smooth points with a unique tangent, Regular and singular loci. Then the local ring ${\mathcal{O}}_{C,p}$ is a discrete valuation ring one dimensional regular local rings are dvrs, A local ring is a nonzero commutative ring with a unique maximal ideal, Discrete valuation rings, 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.

Its maximal ideal is generated by the image of any local equation $l$ of a line through $p$ that is not the tangent line to $C$ at $p$ The Jacobian kernel computes the tangent space, The intrinsic Zariski tangent space; such an $l$ is a uniformising parameter at $p$ Uniformising parameters, embedding dimension and regular local ring. The associated valuation

${\mathrm{ord}}_p\text{\hspace{0.17em}⁣}:\mathrm{Frac}({\mathcal{O}}_{C,p}{)}^{\times }\to \mathbb{Z}$

is the order of vanishing along $C$ at $p$. Its domain is the fraction field of the local DVR, equivalently the function field of the unique irreducible component through the smooth point $p$; when $C$ is irreducible this is $k(C)$. On nonzero local germs, it is positive exactly for the nonzero rational functions on $C$ that vanish at $p$, vanishes exactly on the units of ${\mathcal{O}}_{C,p}$, and for $n\ge 1$ the nonzero germs of order at least $n$, together with zero, form the ideal ${\mathfrak{m}}_p^n$ of ${\mathcal{O}}_{C,p}$. This follows from the normal form $h=u{\pi }^r$ in a DVR Every nonzero fraction is a unit times a power of a uniformiser.

• Existence of non-tangent lines. Through a point of ${\mathbf{P}}^2$ there pass three or more lines; at most the one tangent line to $C$ is excluded, so a non-tangent line through $p$ exists. Its local equation is a linear form with nonzero image in the one-dimensional cotangent space of ${\mathcal{O}}_{C,p}$ (the image is nonzero exactly because the line is not the tangent line), hence generates the maximal ideal of the DVR: writing $l=u{\pi }^r$, its nonzero class modulo $({\pi }^2)$ forces $r=1$ Every nonzero fraction is a unit times a power of a uniformiser.
• Units and vanishing. A germ is a unit of ${\mathcal{O}}_{C,p}$ exactly when it does not vanish at $p$, so ${\mathrm{ord}}_p(u)=0$ for units and ${\mathrm{ord}}_p(h)>0$ for $h$ vanishing at $p$; this is the sense in which ${\mathrm{ord}}_p$ measures the order of vanishing along the curve. The value ${\mathrm{ord}}_p$ is independent of the chosen uniformiser because any two uniformisers differ by a unit.
• Choice. The definition inherits the Axiom of Choice from the DVR and Jacobian-criterion suppliers: the DVR identification and the local-dimensional and classical-variety suppliers inherit it. The rational-point Jacobian-kernel statement itself needs no choice The Axiom of Choice. No additional choice is made here.