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Does anybody know of a good way to figure out if the variety defined by the ideal $I=(a_1y_2-a_2y_1,b_1y_2-b_2y_1)$ in $\mathbb A^2_{y_1,y_2}\times\mathbb P^1_{a_1,a_2}\times \mathbb P^1_{b_1,b_2}$ is Toric?

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    Your scheme has two irreducible components: On one of them, the equation $y_1=y_2=0$ holds, on the other one $a_1 b_2 = a_2 b_1$ holds. Each component is toric. So this isn't a variety (at least, according to Hartshorne) and isn't a toric variety (at least, according to Fulton).2011-02-17
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    I actually hadn't though about it being irreducible or not! Do you know how can I find the components? The in the second component you mention, the equation $a_1b_2-a_2b_1$ is not enough, since this imposes no conditions on the $y$'s and you must have $[y_1,y_2]=[a_1,a_2]$ on $X$ as long as $(y_1,y_2)\neq (0,0)$. Thanks!2011-02-17

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