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?



Is the toric variety associated to this fan a weighted projective space?
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 $t^N$.


