Let $C$ be a degree $d$ irreducible curve in $\mathbb{CP}^{2}$. Can we find maximum number of singular points in $C$?
For $d=2$, I find that there is no singular point on irreducible conic. (If there is a singular point, then consider any line $L$ passes through that singular point and any other point on $C$. If $L$ is not a component of $C$, by Bezout's theorem, we have $2=\sum I_{p}(C, L)\geq 1+2=3$, contradiction. (Here we use the fact that if $p$ is singular point of $C$, then $I_{p}(C, D)\geq 2$ for any curve passes $p$.)
For $d=3$, every irreducible cubic curve can have at most one singular point. If there are two singular points on $C$, consider a line connecting these two singular points. By Bezout's theorem again, $3=\sum I_{p}(C, L)\geq 2+2=4$, contradiction.
How can we generalize these argument? Thanks in advance.