### On the local description of varieties

The degree of a curve is the degree of the defining polynomial. For example, a line is a curve of degree $1$.

Let $F$ be a curve in $\Bbb{A}[X,Y]$, and $P=(a,b)\in F$. Then $P$ will be called a simple point if $F_X(P)\neq 0$ or $F_Y(P)\neq 0$. What does this condition really imply? It means that the function doesn’t locally stop changing as we move away from $x$ and $y$. It is not a local maximum, or minimum (it’s not an extremal point). In this case the line $F_X(X-a)+F_Y(Y-b)=0$ is called the tangent line. Why do we get tangent lines whose equations look like this? This is analogous to the dot product rule for writing down the equation to a plane (or any hyperplane in general). The line is the hyperplane in $\Bbb{A}^2$.

Take an equation $F$, and write it down as $F=F_m+F_{m+1}+\dots+F_n$, where $F_m\neq 0$. Here each $F_i$ is a homogeneous component. Then $m$ is the multiplicity of $F$ at $(0,0)$. What does this mean? As we approach $(0,0)$, the higher degree terms approach $0$ much faster than $F_m$. Hence, $F$ “looks like” $F_m$ near $0$. An analogy would be $0=xy+(xy)^2+(xy)^3$. Near $(0,0)$, the function “looks like” $xy=0$. Hence, it makes sense to call $m$ the multiplicity of the curve at $(0,0)$.

But how do we define the multiplicity of the curve at an arbitrary point on it? Why only $(0,0)$? We can create new variables $x'=(x-a)$ and $y'=(y-b)$, and then re-write the equation. The new equation contains the point $(0,0)$. Now we can follow the same procedure.

As $F_m$ is a function of two variables in this case, it can be factorized into a product of linear polynomials, raised to various exponents. Is it weird that the function looks like a bunch of lines near $(0,0)$? No. Locally, every polynomial, and in fact every function, looks like a linear near a point. Here, because we’re talking about varieties, and hence possibly a product of polynomials, the local description of a polynomial is a bunch of lines. Nothing weird in it at all.

Why are the factors of $F_m$ tangents at $(0,0)$? Are we taking the “local” argument too far? No. It’s a revelation though. I have never seen it before. It makes complete mathematical sense. Say we take $y-x^2=0$. Then the tangent at $(0,0)$ is $y=0$, which is in fact true.