3
$\begingroup$

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?

$\endgroup$
7
$\begingroup$

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 $t^N$.

$\endgroup$
1
  • 1
    $\begingroup$ Replace $t$ by $t^N$, Of course! How did I miss that! Thank you. $\endgroup$ – R_D Jun 8 '16 at 6:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.