Let $F(X_0,X_1,X_2)=-X_1^4+X_1^3X_2+X_0^3X_2$ and $C:=\{F=0\}\subset\mathbb{C}\mathbb{P}^2$. I am asked to find a quartic $D\subset\mathbb{C}\mathbb{P}^2$ subject to the following conditions (here $I(C\cap D,P)$ denotes the intersection index):
- $I(C\cap D,(1:0:0))\geq7$
- $I(C\cap D,(0:0:1))\geq6$
- $I(C\cap D,(0:1:1))\geq2$
- $(2:2:1)\in D$.
- $(1:1:1)\in D$.
I'm not sure how I should proceed to solve this exercise. May be it is important that since $(2:2:1)\in C\cap D$, we get summing all the conditions of intersection: $$7+6+2+1=16=4\times 4$$ which implies by Bezout's theorem that there are no other points in $C\cap D$, and that the $\geq$'s are actually equalities.