# One point in intersection of conic and cubic, intersection multiplicity?

Let $C \subset \mathbb{P}^2$ be the conic defined by $P(x, y, z) = xz + y^2$,and $D \subset \mathbb{P}^2$ be the cubic defined by $Q(x, y, z) = y^2z - x^3 + xz^2$. Show that there exactly one point $p \in C \cap D$. What is $I_p(C, D)$?

• Welcome to MSE, I have a couple of comments about how you could improve your question. First note that $C$ is not an element of $\mathbb{P}^2$, it is a subset of $\mathbb{P}^2$ so you should write $C \subset \mathbb{P}^2$; likewise for $D$. Also, you should provide some context for your question. Do you understand what is being asked? Were you able to make some progress on any of the parts? Also, it is not clear what $\mathbb{P}^2$ is. Is it a projective space over any field or some particular field? You should make sure to introduce any notation that you're using. – Michael Albanese Sep 6 '15 at 22:38

Since $Q - zP = -x^3$, we find that $x = 0$ at any point of $C \cap D$. If $x = 0$, then $P = 0$ implies $y = 0$. We find that $p = [0, 0, 1]$ is the unique point of intersection. By Bézout's theorem, since $p$ is the only point in $C \cap D$, we have that $I_p(C, D) = 2 \cdot 3 = 6$.