w \[ \begin{aligned} \phi(x, t) &= (x,\; xt), \quad \det J_\...

Prompt

w \[ \begin{aligned} \phi(x, t) &= (x,\; xt), \quad \det J_\phi = x, \\ G(x, t) &= (F \circ \phi)(x, t) = \left(x + \frac{t^2}{2},\; xt + \frac{t^3}{3}\right), \\ $J_G &= \begin{pmatrix} 1 & t \\ t & x + t^2 \end{pmatrix}, \quad \det J_G = x, \\ \det J_F &= \frac{\det J_G}{\det J_\phi} = \frac{x}{x} = \mathbf{1}, \\ F(x, y) &= \left(x + \frac{y^2}{2x^2},\; y + \frac{y^3}{3x^3}\right), \\ 8F_1^3 - 9F_2^2 &= 8x^3 + 3y^2, \\ t^3 - 6Xt + 6Y &= 0 \implies \operatorname{Disc}_t = 108\left(8X^3 - 9Y^2\right) \quad (3:1 \text{ coverage}). \end{aligned} \]