Consider the complete fan $\Delta$ in $\mathbb R^2$ with edge vectors $$ v_1=e_1,\qquad v_2=-a_1e_1+a_2e_2, \qquad v_3=-b_1e_2-b_2e_2\,, $$ where $a_1,a_2$ and $b_1,b_2$ are respectively relatively prime positive integers. Then the corresponding toric variety $X(\Delta)=\mathbb C^3\setminus\{0\}/\mathbb C^*$ where the $\mathbb C^*$ action on by pointwise multiplication under the identification $$\mathbb C^*\cong\left\{\left(t^{\frac{a_1}{a_2}+\frac{b_1}{b_2}},t^{\frac{1}{a_2}},t^{\frac{1}{b_2}}\right)\, \mid\, t\in\mathbb C^*\right\}$$

Is there a specific name for such a toric variety? Is this toric variety a weighted projective space of some kind?


Let $N:=\text{lcm}(a_2,b_2)$, $A_2:=\frac{N}{a_2}$, $B_2:=\frac{N}{b_2}$. Then your toric variety fits the definition of the weighted projective space $\mathbf P(a_1A_2+b_1B_2,A_2,B_2)$. Just replace $t$ by $t^N$.

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    $\begingroup$ Replace $t$ by $t^N$, Of course! How did I miss that! Thank you. $\endgroup$ – R_D Jun 8 '16 at 6:34

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