equations for a bidouble cover I am now trying to construct a $Z_2\times Z_2$-cover over $\mathbf{P}^n$. From the paper of Pardini, we need line bundles $L_1$, $L_2$, $L_3$ and divisors $D_1$, $D_2$, $D_3$ which satisfies the following 6 relations.
$$2L_i\equiv D_j+D_k\quad \textrm{and} \quad L_i+L_j\equiv L_k+D_k$$
However, the paper introduces 'reduced' data, which only need the following 2 relations.
$$2L_1\equiv D_2+D_3\quad \textrm{and} \quad 2L_2\equiv D_1+D_3$$
For an example, let $n=2$ and take $D_i$ to be the locus $X_i=0$.
Then, for fiber coordinates $y_1$, $y_2$, and local coordinates $x_2$, $x_3$ (on $U_1$ where $X_1\neq 0$), if I am right, the equation of the bidouble cover can be written by
$$y_1^2=x_2x_3,\quad y_2^2=x_3.$$
By calculating its Jacobian, the bidouble cover should be singular at $(x_2,x_3,y_1,y_2)=(0,0,0,0)$, which is false.
So I am wondering that is it wrong to think of such local equations from the 'reduced' data. 
 A: As abx note in the comments, you miswrote the equations. Still the singular point remains. 
The point is that the "reduced" data works well algebraically, as indeed you can deduce $L_3$ from the other data. But if you try to "reduce" analogously the equations you are eliminating some variables and that correspond geometrically to a projection which, in this case, is just a birational morphism, whose image has a singular point.
I give you here what are the "best", in my opinion, equations for this bidouble cover, determinantal equations that can be used to describe efficiently most bidouble covers. I apologize in advance for possible mistakes, because I'm in a rush, having exams to do right now. I'll check this post better in few hours.
The bidouble cover you consider can be see as a determinantal locus in ${\mathbb P}^5$, by taking variables $x_1,x_2,x_3,y_1,y_2,y_3$ and considering the locus given by the condition that the symmetric matrix
$$
\begin{bmatrix}
x_1&y_3&y_2\\
y_3&x_2&y_1\\
y_2&y_1&x_3
\end{bmatrix}
$$
be of rank $1$. 
You can recognize the three equations $y_i^2=x_ix_j$ giving a  tridouble cover, the three involutions being given by $y_i\mapsto -y_i$, that has two irreducible components, and the remaining three equations $x_iy_i=y_jy_k$ determine one of the two components, invariant by three involutions, obtained by changing sign simultaneously to two of the $y_j$. 
