Let $C$ be given by the zeros of the polynomial $$f = X^2 -XY + Y^3$$ find the horizontal tangent lines of $C$
The way I interpret "horizontal line" is that it is given by the direction $(1,0)$. So let $F = X_0X_1^2-X_0X_1X_2+X_2^3$ be the projective completion of $C$, $\overline{C}$. Then I have to find the tangent lines that go through the point $(0:1:0)$. Now, I know that, if $\Sigma = \{ b\in C \mid a\in \mathbb{T}_bC \}$, then $$C \cap P(a,C) = \mathrm{Sing}(C) \cup \Sigma$$ where $P(a,C)$ is the polar curve of $C$ with respect to $a$. In this case, $a = (0:1:0)$, and the polar curve is given by $$P(a,C) = V(F_0) = V(X_0(X_1-X_2))$$ Now if $X_0 = 0$, then $X_2 =0$ and we get the point $(0:1:0)$. If $X_1 = X_2$, we get the point $(1:0:0)$. And there are no more points in $P(a,C)\cap C$. Of those two, $(1:0:0)$ is singular, so $$\Sigma = \{ (0:1:0) \}$$
Is this correct? How can I interpret this result?