dimension of invertible sheaves quadric surface (Riemann-Roch)

Question

On the quadric surface $xy = zw$ in $\mathbb{P}^3$ the divisor class group is isomorphic with $\mathbb{Z} \oplus \mathbb{Z}$. I want to see what the dimension of $\mathcal{L}(D)$ is for a divisor $D$. I have an idea about it, but I don't find any references about this. My idea is the following. Let $D$ be a divisor of type $(a,b)$, thus given by the union $l^a \cup m^b$ for the lines $l = \mathbb{P}^1 \times P, m = Q \times \mathbb{P}^1$ for some points $P,Q \in \mathbb{P}^1$. Then $\mathcal{L}(D)$ consists of all rational functions $f$ on $\mathbb{P}^1 \times \mathbb{P}^1$ such that $(f) \geq -a l - bm$. Let the coordinates on $\mathbb{P}^1 \times \mathbb{P}^1$ be given by $(x_0,x_1) \times (y_0,y_1)$. Then the space $\mathcal{L}(D)$ is generated by all rational functions of the form $F/x_0^a + G/y_0^b$ for $F(x_0,x_1),G(y_0,Y-1$ homogeneous of degree $a$ and $b$, thus $l(D) = \dim_k \mathcal{L}(D) = \binom{a+1}{1} \binom{b+1}{1} = (a+1)(b+1)$ if $a \geq 0, b \geq 0$ and $l(D) = 0$ if $a < 0$ or $b< 0$. Then $K-D$ is of type $(-2-a,-2-b)$ and thus $l(K-D) = (-1-a)(-1-b) = (a+1)(b+1)$ if $a \leq -1, b \leq - 1$ and $l(K-D) = 0$ if $a > -1$ or $b > -1$. Does anyone with more background about this can confirm this or give a reference where to look? Thank you!