I'm failing to see how exactly is the tangent cone at a singular point on a curve picking out all the different tangent lines through this singular point (say the origin in $\mathbb{A}^2$)?

Could someone explain this, or at least redirect me to a source I could read about? I tried to look online, but at most places they are just taking this as a known fact!



Initial remark. Suppose someone gives you a continuous function of two variables, and asks you to calculate the Taylor series in $(0,0)$. If we happen to notice that the first derivatives vanish at the origin, we will call it a critical point. The idea of the tangent cone is that we are doing exactly this: taking the Taylor series of the simplest continuous function: a polynomial. And if the first derivatives vanish, we call $(0,0)$ a singular point.

We have a curve $X=V(f)\subset \mathbb A^2$ passing through the origin. Even if the origin is a nonsingular point of the curve, you do have a tangent cone at the origin: it is also called the tangent line! (and this is precisely encoded in the corresponding first terms of the "Taylor series" of $f$).

However, the tangent cone is constructed starting from the leading form of $f\in k[x,y]$, which is the homogeneous form $\tilde f$ of smallest degree appearing in the decomposition of $f$; it is again in two variables. For instance, the leading form of $f=2x+y-8y^2x$ is $\tilde f=2x+y$, the term of smallest degree. The tangent cone is the zero set $V(\tilde f)$. So the origin is nonsingular if and only if $f$ has leading form of degree one. In that case, $V(\tilde f)$ is exactly the tangent line.

In general, the leading form will be a product of linear factors, each appearing with some exponent. So the scheme $V(\tilde f)$ is a union of lines, where some of them are possibly nonreduced.

If $f$ starts, say, with degree $2$, then for instance $(0,0)$ will be a node in case the (degree $2$) leading form is a product of two distinct linear forms, i.e. something of the kind $$(ax+by)\cdot (cx+dy).$$ If, instead, the leading form has the shape $(ax+by)^2$, something else happens (example: the cusp $f=y^2-x^3$, whose tangent cone which is a double line).

Reference. All this is beautifully explained in Mumford's The red book of varieties and schemes.

  • $\begingroup$ I guess my question then $\bf{really}$ is how should I go about understanding that when I write $f=f_0+f_1+...+f_d$ where $f_i$ are homogeneous polynomials of degree $i$ in $x,y$, then why is the smallest such $f_r$ picking out all these different tangent lines at the singular point? Because, I think, I can see how it is working, I just don't seem to understand why. I'm not sure if I'm making sense? Thanks for the reference, by the way, I will definitely check that out! $\endgroup$ – V-B Mar 18 '14 at 23:13
  • $\begingroup$ You call them tangents because you are selecting the least degree in the homogeneous decomposition of $f$. You have them all just by definition (you take the entire homogeneous part). You actually see them one by one writing down the product decomposition of $\tilde f$ into linear factors. Sorry for being unclear on that. $\endgroup$ – Brenin Mar 18 '14 at 23:24
  • $\begingroup$ My question might be more ignorant than I thought. Here is my confusion: Say that the leading form of $f$ factors into linear factors. Then, these two lines seem to be precisely the tangents at the singular point, of the two branches of $f$ respectively. For example take $f=y^2-x^2-x^3$. The leading form factors as $(x-y)(x+y)$ and these are precisely the tangent lines through the origin of the two branches of $f$. I don't see how this is happening in general. How are the other terms irrelevant? I might be missing something more fundamental. $\endgroup$ – V-B Mar 18 '14 at 23:43
  • $\begingroup$ It seems to me, like somehow we are writing $f$ as a product of some formal power series, and then these are defining each branch of $f$, and then in turn the linear term of these formal power series is representing the 'tangent' of the corresponding branch through this singular point of $f$. But, also, I'm not sure if we can always write $f$ as a product of these formal power series. $\endgroup$ – V-B Mar 18 '14 at 23:46
  • $\begingroup$ As for your first comment: I do not get what you mean when you say the other terms are "irrelevant". Also, I do not see power series hidden as factors of the polynomial $f$, but it is probably something I am missing myself. Indeed, the study of singularities is very much related to power series (maybe, those you mention), in particular to the completions of the local rings at the singular points. Sorry for not being able to say more on this. $\endgroup$ – Brenin Mar 19 '14 at 20:03

For anyone interested I will write a reference for the 'answer' to this question: It turns out that this is a problem in Hartshorne, namely Problem 5.14 (b) of Chapter 1. I will write the statement here, and maybe at some point post a solution as well.

If $f=f_r+f_{r+1}+\cdots \in k[[ x, y]]$, and if the leading form $f_r$ factors as $f_r=g_sh_t$, where $g_s,h_t$ are homogeneous of degrees $s$ and $t$ respectively, and have no common linear factors, then there are formal power series $$g=g_s+g_{s+1}+\cdots$$ $$h=h_t+h_{t+1}+\cdots$$ in $k[[x,y]]$ such that $f=gh.$


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