Its description is a little bit complicated but it would be great if anyone can give an example. I tried to construct it as a toric variety (See the previous question) but did not succeed.
I am looking for a projective threefold $X$ with $H^1(\mathcal O_X) = 0$ such that it has only three ordinary double points $p_1, p_2, p_3$ as its singularities. Its anticanonical divisor $-K_X$ is linearly equivalent to $$Y_{12}^1 + Y_{23}^1 + Y_{31}^1 + Y_{12}^2 + Y_{23}^2 + Y_{31}^2,$$
where $Y_{ij}^k$'s are divisors of $X$, isomorphic to $\mathbb P^2$. Each $Y_{ij}^k$ goes through $p_i, p_j$. The triangular bipyramid (click the word for its image) can describe how $Y_{ij}^k$'s intersect with another. The faces of the triangular bipyramid corresponds to $Y_{ij}^k$'s. The edges describe how $Y_{ij}^k$'s intersects. If two of $Y_{ij}^k$'s intersect, then they intersect transversely along a line of degree one. Note that vertices of valence 4 correspond to the points $p_1, p_2, p_3$. For fixed $k$, $Y_{12}^k, Y_{23}^k, Y_{31}^k$ meet at a point, corresponding to each of the other two vertices.
To Prof. Lev Borisov (if you take a look at this post): In the previous post, you suggested that I presumably can make it as a toric DM stack. I know little about toric DM stack but I would like to ask if such 'a toric DM stack' would have the above properties.
Thanks for your attention.
