ag.algebraic geometry - Looking for a 3-fold with some property?

There exists such variety which is singular with Gorenstien singularities.

Let $S$ be a del Pezzo surface of degree $d$ and let ${mathcal L} = O_S(−K_S)$.
Consider the $mathbb P^1$ -bundle
$mathbb{P} = mathbb{P}_S (mathcal{O}_S bigoplus mathcal{L})$.

Now the variety $X$ can be constructed as a contruction of a zero divisor. The map $mathbb{P}to X$ given by the linear system
${mathcal{O}}_{mathbb{P}}(n), quad n ≫ 0.$

It contracts the negative section. Since $−K_{mathbb P} ∼ O_{mathbb P}(2),$
the variety $X$ is a Fano threefold of index $2$ and degree $8d$ with canonical Gorenstein singularities. For $S = {mathbb P}_2$ we have $−K^3_X = 72$ and
$X ≃ mathbb{P}(3, 1, 1, 1)$ is a weighted projective space.