# Intersection numbers from matrix of divisors and relations

There is a specific procedure to "read-off" or "compute" intersection products of divisors and curves, and of divisors and exceptional divisiors, from the so called matrix of relations (I don't know if this a standard name) which is schematically represented by $(P | Q)$ where the rows of $P$ are the coordinates of divisors in the toric fan, and the columns of $Q$ are the Mori generators.

The particular example I have in mind is that of the blow-up of $\mathbb{C}^3/\mathbb{Z}_6$ as described on page 23 of the review "The Geometer's Toolkit to String Compactifications" by Sussane Reffert (arXiv:0706.1310). In this case, the pertinent matrix is

$$(P|Q) = \begin{pmatrix}D_1 & 1 & -2 & 1 & | & 0 & 1 & 0\\ D_2 & -1 & -2 & 1 & | & 0 & 1 & 0\\ D_3 & 0 & 1 & 1 & | & 0 & 0 & 1\\ E_1 & 0 & 0 & 1 & | & 1 & 0 & -2\\ E_2 & 0 & -1 & 1 & | & -2 & 0 & 1\\ E_3 & 0 & -2 & 1 & | & 1 & -2 & 0 \\ & & & & & C_1 & C_2 & C_3 \end{pmatrix}$$

So, as Reffert says

From the rows of Q, we can read off directly the linear equivalences: $$D_1 \sim D_2, \quad E_2 \sim -2E_1-3D_3, \quad E_3\sim E_1-2D_1+2D_3$$

Question 1: Am I correct in understanding that these equivalences are really of the corresponding cohomology classes?

Question 2: It is trivial to see that $D_1 \sim D_2$ from the first two rows of $Q$. However, while the rest are easy to verify, they are not unique relations. For instance, at the level of the classes, I see that $D_3 \sim -2D_1 + E_1 + 3E_2 -E_3$, so is this a matter of choice?

I understand that from the $Q$ matrix, one can read off the intersection products of divisors with curves $C_1, C_2, C_3$, that is, things like $D_i \cdot C_j$ and $E_i \cdot C_j$ for $1 \leq i, j \leq 3$: those are just the corresponding matrix entries of $Q$. However, the author remarks

We know that $E_1 \cdot E_3 = 0$

Question 3: How can one infer this fact ($E_1 \cdot E_3 = 0$) from the matrix alone? (I know this is true from the toric diagram or the dual toric diagram. But this matrix is supposed to encode everything about the fan, so in particular this should be inferrable from the matrix itself.)

Also, the author goes on to say the following:

From the linear equivalences between the divisors, we find the following relations between the curves $C_i$ and the seven compact curves of our geometry:

$$C_1 = D_1 \cdot E_2 = D_2 \cdot E_2$$ $$C_2 = E_2 \cdot E_3$$ $$C_3 = D_1 \cdot E_1 = D_2 \cdot E_1$$ $$E_1\cdot E_2 = 2 C_1 + C_2$$ $$D_3\cdot E_1 = 2 C_1 + C_2 + 4C_3$$

Question 4: How does one get these relations from the matrix $(P|Q)$?

I am sure I am missing something really simple, but I couldn't get my head around this by reading just these notes. I worked out a few examples from the Mirror Symmetry book by Hori et al. but the examples that I could find there were for 2 complex dimensions, where divisors are curves.