If $\mathcal{F}$ is a coherent sheaf on a smooth toric variety $X$ then multigraded commutative algebra can be used to compute the cohomology groups $H^i(X, \mathcal{F})$.
Indeed if $B$ is the irrelevant ideal of $X$ then the cohomology group $H^i(X, \mathcal{F})$ can be realized as the degree zero piece of the multigraded module $Ext^i(B^{[l]}, F)$ for sufficiently large $l$; here $B^{[l]}$ denotes the $l$th Forbenius power of $B$ and $F$ is any multigraded module whose corresponding sheaf on $X$ is $\mathcal{F}$.
Given the fan of $X$ and $F$ a sufficiently large power of $l$ can be determined effectively. We refer to sections 2 and 3 of the paper "Cohomology on Toric Varieties and Local Cohomology with Monomial Supports" for more details.
In this example, we consider the case that $X = \mathbb{P}^1 \times \mathbb{P}^1$ and $F = \mathcal{O}_C(1,0)$ where $C$ is a general divisor of type $(3,3)$ on $X$. In this setting, $H^0(C,F)$ and $H^1(C, F)$ are both $2$-dimensional vector spaces.
We first make the multi-graded coordinate ring of $\mathbb{P}^1 \times \mathbb{P}^1$, the irrelevant ideal, and a sufficentily high Frobenus power of the irrelevant ideal needed for our calculations. Also the complex $G$ below is a resolution of the irrelevant ideal.
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We next make the ideal, denoted by $I$ below, of a general divisor of type $(3,3)$ on $\mathbb{P}^1 \times \mathbb{P}^1$. Also the chain complex $F$ below is a resolution of this ideal.
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To use hypercohomology to compute the cohomology groups of the line bundle $\mathcal{O}_C(1,0)$ on $C$ we twist the complex $F$ above by a line of ruling and then make a filtered complex whose associated spectral sequence abuts to the desired cohomology groups.
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The cohomology groups we want are obtained as follows.
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