In Fulton's Algebraic Curves, beginning of chapter 3, there is a quick discussion about multiple points and tangent lines.
He gives many examples of affine curves, for instance:
He then says that the lowest terms $Y^2-X^2$ in $D$ and $3X^2Y-Y^3$ in $E$, in his words, "pick out those lines that can best be called tangent to the curve at $(0,0)$"
In deed, after drawing the lines defined by $Y^2-X^2=(Y-X)(Y+X)=0$ and by $3X^2Y-Y^3=Y(\sqrt{3}X-Y)(\sqrt{3}X+Y)=0$, I've checked that they fit perfectly in the picture as tangent lines.
I don't understand why this happens. In general, why should I suspect that for a curve $F=F_m+…+F_d$ (where $F_i$ are homogeneous of degree $i$), the term $F_m$ should give me all the tangent directions?
Best Answer
We have a curve $C$, which is the zero set of some equation $F$ (assume $F(0,0)=0$, so $C$ passes through the origin).
If we have a vector $v\in k^n$, we can talk about the line through the origin in the direction $v$, $\ell = \{tv : t\in k\}$, and we can discuss what $F$ restricted to that line looks like.
Well, if we are working in the plane, $v = (x,y)$ so the line is $(tx,ty)$, and $$F|_L(t)=F_m(tx,ty) + F_{m+1}(tx,ty)+\cdots +F_n(tx,ty).$$
Since each of the forms $F_i$ is homogeneous, we can pull the $t$s out to get $$F|_L(t) = F_m(x,y)t^m+F_{m+1}(x,y)t^{m+1}+\cdots +F_n(x,y)t^n.$$
Then if we ask about the multiplicity of the zero of $F|_L$ at $0$, we find that it is the first $i$ such that $F_i(x,y)\ne 0$. For most choices of $x$ and $y$, we will have $F_m(x,y)\ne 0$, so for most lines, the multiplicity of the zero of $F|_L$ at the origin is $m$, the same as the multiplicity of $F$ at the origin. However for lines in directions $(x,y)$ where $F_m(x,y)=0$, the multiplicity goes up. These directions correspond to tangent directions, since when you travel in a tangent direction, the function defining an implicit surface should change more slowly or have an extra zero (since the tangent directions should intuitively correspond to infinitesimal movements that change the value of the function the least).
Then, in particular, in two dimensions, we can use the fact that every homogeneous polynomial in two variables over an algebraically closed field splits into linear factors. Each of these factors will correspond to one of the direction vectors $(x,y)$ for which $F_m(x,y)=0$. Thus, we get that the tangent directions in two dimensions are determined by the linear factors of the lowest degree form.
To focus on your specific question, the reason the low degree form is what matters is that the lowest degree form has to be zero for the multiplicity of a zero to be greater than $m$.