Consider $P=\mathbb{P}^3\times \mathbb{P}^1$ with coordinates $([x_0:x_1:x_2:x_3],[u:v])$ and let $\varepsilon:X\to P$ be the blow-up of $\mathbb{P}^2\cong A=(x_3=v=0)\subseteq P$, of pure codimension 2.
We know that
- $K_P=\mathcal{O}_P(-4,-2)$, where $\mathcal{O}_P(a,b):=\operatorname{pr}_1^*\mathcal{O}_{\mathbb{P}^3}(a)\otimes \operatorname{pr}_2^*\mathcal{O}_{\mathbb{P}^1}(b)$, and
- $K_X \cong \varepsilon^*K_P + E$, where $E\cong \mathbb{P}^2 \times \mathbb{P}^1$ is the exceptional divisor.
My question: Is $-K_X$ ample ?
By Kleiman's ampleness criterion we must show that $K_X\cdot C < 0$ for every irreducible curve in $X$.
If $C$ is not contained the exceptional divisor $E$ then it follows from the fact that $-K_P$ is ample.
If $C$ is contained in $E$ then it is numerically equivalent to $a\ell_1+b\ell_2$, $a,b\geq 0$, where $\mathbb{P}^1\cong \ell_1 = \{\text{point}\}\times \mathbb{P}^1$ and $\mathbb{P}^1\cong \ell_2 \subseteq \mathbb{P}^2 \times \{\text{point}\}$ are the generators of $\operatorname{NE}(E)$.
So the question is (if there are no mistakes): how to compute $E\cdot \ell_i$ ?
Thank you very much !