What Is Intersection Multiplicity and How Is It Calculated?

[Fermat1]

Define the multiplicity of $f$ at $p$ and the interesction multiplicity of $f,g$ at $p$.

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$.

If we let $f=x^2-3x+y^2$ and $g=x^2-6x+10y^2$ my book simplifies $<f,g>$ to $<x,y^2>$. Then my book says that $\mathcal{O}_{\mathbb{A}_k^2,O}/<x,y^2>$ is isomorphic to $\mathbb C<1,y>$. What is meant by the notation $\mathcal{O}_{\mathbb{A}_k^2,O}$ and how do they get this isomorphism? Thanks

 

[jvicens]

The multiplicity of a curve $A$ at a point $p$ is the highest power of the local parameter at $p$ that divides the defining equation of $A$. In other words, it is the number of times the curve "touches" the point $p$.

The intersection multiplicity of two curves $A$ and $B$ at a point $p$ is the multiplicity of the product of their defining equations at $p$. In other words, it is the number of times the two curves intersect at the point $p$.

In this case, the intersection multiplicities at any point of intersection of $A$ and $B$ can be determined by looking at the ideal generated by their defining equations, which in this case is $<x,y^2>$. The notation $\mathcal{O}_{\mathbb{A}_k^2,O}$ refers to the local ring of the affine plane $\mathbb{A}_k^2$ at the origin $O$. This is the ring of all regular functions on $\mathbb{A}_k^2$ that are defined at the origin.

The isomorphism between $\mathcal{O}_{\mathbb{A}_k^2,O}/<x,y^2>$ and $\mathbb C<1,y>$ can be obtained by using the fact that the quotient of a polynomial ring by an ideal is isomorphic to the ring of all polynomials in the variables not contained in the ideal. In this case, we are essentially "eliminating" the variable $x$ from the local ring, leaving us with a polynomial ring in the variable $y$ over the field $\mathbb C$.