Line multiplicities at a cusp

Given: AC The Axiom of Choice, an algebraically closed field $k$, the curve $C=V(y^{2}z-x^3)$ with the affine chart $z=1$, affine equation $f=y^2-x^3$, the cusp $p=(0,0)$, and $O=k[x,y{]}_{(x,y)}$.

The homogeneous equation $y^{2}z-x^3$ is primitive and linear in $z$ over $k[x,y]$, so it is irreducible by Gauss lemma over a UFD, Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes. Its dehomogenisation is square-free, so $C$ is a plane projective curve; in the chart $z=1$ the point $p$ has $m_p(C)=2$ and the lowest-degree part of $f$ is $y^2$, so the tangent cone is the double line $V(y)$ Plane projective curves and their components, Multiplicity of a plane curve at a point, Tangent cone and tangent lines at a point.

The local intersection multiplicity is the length of $O\mathrm{/}(f,g)$ for a local equation $g$ of the second curve, and it is computed by $I_p(C,D)={\mathrm{\ell }}_O(O\mathrm{/}(f,g))$ Local intersection multiplicity of two plane curves. Simplicity: $O\mathrm{/}(y,x^3)$ has $k$-basis the classes of $1,x,x^2$, and $O\mathrm{/}(x,y^2)$ has $k$-basis the classes of $1,y$ Each quotient is respectively $k[x{]}_{(x)}\mathrm{/}(x^3)$ or $k[y{]}_{(y)}\mathrm{/}(y^2)$; the descending-power flag has simple residue-$k$ factors, hence lengths three and two Composition series and length of a module, Finite local length exactly when no common local branch.

The point is a double point, so the product bound gives $I_p(C,L)\ge 2$ for every line $L$ through $p$, with equality exactly when the tangent cones are separated; the tangent line $V(y)$ shares its (double) tangent direction with the cusp, so there the value is at least $3$ Intersection multiplicity dominates the product of multiplicities, with equality for separated tangent cones.