Intersections on Surfaces

The quadratic and an axis.

(Taken from my notes on intersection theory)

Let $X= \mathbb P^2$, $C$ be the vanishing set of the global section $x^2-yz$ of $\mathcal O(2)$, and $D$ be the vanishing set of the global section $y$ of $\mathcal O(1)$ (here by vanishing of $f$ we mean the set of all points $P$ where $f\in \mathfrak m_P$). The intersection $C\cap D$ contains all the points where $y$ and $x^2-yz$ are in $\mathfrak m_P$. Consider the affine patch $\{z\neq0\}$, and suppose that $C\cap D$ has a point $P$ outside that open set. Clearly the point $P=[0:0:1]$ in this patch is in $C\cap D$. We seek to compute the intersection multiplicity at this point. For this it will suffice to work in the affine plane, with the equations $x^2-y = 0$ and $y = 0$. Then we compute $\mathcal O_{X, P} = k[x, y]_{(x, y)}$ and

and so $(C\cdot D)_P=2$. This confirms our intuition from calculus that the multiplicity of the quadratic’s zero is 2. But are there no other points in the intersection? There are no other intersections in this affine patch, so it remains only to check the points at infinity. If $z\in \mathfrak m_P$, then $x^2-yz\in \mathfrak m_P$ implies $x\in\mathfrak m_P$ by primality, so if $y\in \mathfrak m_P$ we have that $x, y, z\in \mathfrak m_P$. But $(x, y, z)\not\in \mathop{\mathrm{Proj}}k[x, y, z]$ as we exclude the irrelevant ideal, so there is no intersection at infinity and we have verified Bézout’s theorem in this case.