I am trying to read Dynamics of Automorphisms of Compact Complex Surfaces by Serge Cantat, and I am confused by his example of surfaces of degree (2, 2, 2) in section 2.4.6. The setup is the following:
Let $X$ be a smooth surface embedded in $\mathbb P(\mathbb C)^1 \times \mathbb P(\mathbb C)^1 \times \mathbb P(\mathbb C)^1$. Let $\pi_1 : X \to \mathbb P(\mathbb C)^1 \times \mathbb P(\mathbb C)^1$ be the projection given on $X \cap \mathbb A^3(\mathbb C)$ by $(x, y, z) \mapsto (y, z)$, and let $\pi_2, \pi_3$ be the analogous projections forgetting $y$ and $z$, respectively. Then each $\pi_i$ is a ramified cover of degree $2$, so there is an associated involution $s_i : X \to X$ that satisfies $\pi_i \circ s_i = \pi_i$. Let $\sigma_i : X \to \mathbb P(\mathbb C)^1$ be projection onto the $i$'th coordinate. Let $[C_i] \in NS(X)$ be the fundamental class of the fiber over a generic point of $\mathbb P(\mathbb C)^1$. Let $NS_X \subset NS(X)$ be the subgroup generated by $[C_1], [C_2], [C_3]$. According to Cantat,
One easily checks that the three involutions $s_i^*$ preserve the space $NS_X$; on $NS_X$, the matrix of $s_1^*$ in the basis $([C_1], [C_2], [C_3])$ is equal to $$ \begin{pmatrix} -1 & 0 & 0 \\ 2 & 1 & 0 \\ 2 & 0 & 1 \end{pmatrix} $$
I cannot figure out where this matrix comes from. I tried reading the section of Hartshorne on divisors, and there were some results that seemed like they could be relevant, but there is a significant amount of material to digest, and I can't tell if it's even the right direction to be looking. What machinery is being used to do this calculation?
This is explained a bit more in the proof of Theorem 3.3 of
Cantat, Oguiso: "Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups". Amer. J. Math. 137 (2015), no. 4, 1013–1044.
The basic idea is to think of $X$ as defined (in the affine chart you specify) by a quadratic equation
$$a(y,z)x^2+b(y,z)x+c(y,z) = 0$$
where $a,b,c$ are quadratic polynomials in $y$ and $z$; then the involution $s_1$ interchanges the roots of this quadratic. So if $x$ is a root of this equation we get
$$ x \cdot s_1(x) = \frac{c(y,z)}{a(y,z)}$$
To conclude, you have to homogenise appropriately and observe that the divisor of zeroes on the left hand side gives you a representative of $[C_1] + s_1([C_1])$, while the divisor of zeroes on the right-hand side gives you a representative of $2[C_2]+2[C_3]$.