Explicitly calculating the tangents to a projective curve

Question

In Fulton's algebraic curves, the definition of a line tangent to a projective curve in $\mathbb{P}^2$ is following:

We can define a line $L$ to be tangent to a (projective) curve $F$ at $P$ if $I(P,F \cap L) > m_P (F)$.

A few more definitions: $I(P,F\cap L)$ is the projective intersection between $F$ and $L$ at $P$ defined to be $\text{dim}_k(\mathcal{O}_P(\mathbb{A}^2)/(F_*, L_*))$, where $\mathcal{O}_P(\mathbb{A}^2)$ is the local ring of the space $\mathbb{A}^2$ at $P$, and $F_*$ and $G_*$ is the dehomogenization of $F$ and $G$ with respect to a coordinate of $P$ that is nonzero (This is not exactly the one written by Fulton, I use an identification shown in this question to make things clear). $m_p(F)$ is the multiplicity of $F$ defined to be the multiplicty of $F_*$ ($F_*$ is defined like above).

Now, my question is,

1. given a projective curve, what is the general procedure to calculate its tangent lines at a point?

In this question, there is a comment that

one approach is to find the tangent line in an affine chart containing a given multiple point, and then take the projective closure (i.e. homogenize the equation of the line).

2. why is it a valid procedure to find the tangents defined by Fulton as I showed above?

In fact, I always have trouble getting a sense of what a projective curve looks like and how one should manipulate them. I have once tried to treat "$\mathbb{P}^2$ as a manifold". Then to get the information (like tangents) of the projective curve at a point, we first identify the curve around the point to its representation in one of the affine chart, calculate the tangent of the identified curve, the "pushforward" the tangents back to the projective setting. But projective curves are not necessarily parametrized so this is in some sense fundamentally different from how one can do in differential geometry.

Edit: Thanks for Daniel to point out in fact the (implicit) answer is already in my question. I will (explicitly) answer both my questions together. WLOG, assume $P = [0:0:1]$; to find the tangents to a projective curve $F$, based on definitions, we need to find the projective lines $L = L(X,Y,Z)$ such that

\begin{aligned} I((0,0), F(X,Y,1)\cap L(X,Y,1))) &= \text{dim}_k(\mathcal{O}_P(\mathbb{A}^2)/(F(X,Y,1),L(X,Y,1))) \\& = I(P,F\cap L)\\ &> m_p(F)\\ &= m_{(0,0)}(F(X,Y,1)) \end{aligned} Now there are two connecting pieces needed.

1. by problem 3.19 on Fulton, an affine line $L'$ is tangent to an affine curve $F'$ at a point $P'$ if and only if $I(P',F' \cap L') > m_{P'} (F')$

2. by proposition 5 on section 2.6 on Fulton, if $F\neq 0$ and $r$ is the highest power of $X_{n+1}$ that divides $F$, then $Z^r(F_*)^* = F$. So in particular, for a line in projective space, $(L_*)^* = L$.

Thus, essentially, after finding the tangent lines of $F(X,Y,1)$ in the usual affine case, homogenizing them precisely give all the tangents of $F$ in the projective case.

As for my "manifold treatment", if the projective curve can be smoothly parametrized, then its tangent at a point is exactly given by the tangent of the "pushforward" of the tangent of the affine identified curve as I described above.

Answer 1

Assume for convinience that P is the origin in some chart and write the equation of $F_*$ as: $$ F_*=\sum_{i,j}a_{ij}x^iy^j, $$ The tangent lines will be the zeros of the polynomial: $$ T=\sum_{i+j=m} a_{ij}x^iy^j, $$ Where $m$ is the multiplicity of $F$ at $P$.

Notice that if $L=bx+cy$ divides $T$, then $I(P,F\cap L)>m$.

Answer 2

You don't really need to work on an affine chart.

Let $\cal{C}\subset\mathbb{P}^2$ be the curve given by the equation $$ F(X_0,X_1,X_2)=0 $$ where $F$ is a form of degree $d>0$. Let $P\in\cal{C}$ be a smooth point, i.e a point such that $$ (F_0(P),F_1(P),F_2(P))\neq(0,0,0), $$ where $F_i=dF/dX_i$.

Then the tangent line at $P$ is the projective line $$ F_0(P)X_0+F_1(P)X_1+F_2(P)X_2=0. $$