Let $X$ be a surface as in the title. Rick Miranda said that $X$ is a Steiner cubic in $\mathbb{P}^4$, and the cover map is projection. Invariants of $X$ can be computed directly, $p_g(X)=0,K^2_X=8,e(X)=4$.

My question is,

Question:What is a Steiner cubic? Why $X$ is a Steiner cubic?

The followings are what I know: Given $a,b,c,d\in H^0(\mathbb{P}^2,\mathcal{O}(1))$, $X$ is locally defined by \begin{cases} F(z,w)=z^2-aw-bw-2(a^2-bd) \\ G(z,w)=zw+dz+aw+ad-bc\\ H(z,w)=w^2-cz-dw-2(d^2-ac) \end{cases} Does, in fact, $X$ globally defined as an intersection of three quadrics in $\mathbb{P}^4$?