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.