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. 



equations for a bidouble cover
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 $y\_j$. 


