$$ \begin{aligned} \mathbf{F} = (\mathbf{P}, \mathbf{Q}): \m...

Prompt

$$ \begin{aligned} \mathbf{F} = (\mathbf{P}, \mathbf{Q}): \mathcal{S} \to \mathbb{C}^2 \quad : \quad & \begin{cases} \mathcal{S} = \left\{ (x, y, w) \in \mathbb{C}^3 \;\middle|\; y = w^2 + \dfrac{32}{9} x^{15} w^9 \right\} \cong \mathbb{C}^2_{(x, w)} \\[10pt] \mathbf{P}(x, w) = x + x^{16} w^7 \\[6pt] \mathbf{Q}(x, w) = w^2 \end{cases} \\[12pt] \wedge\quad & \mathrm{d}\mathbf{P} \wedge \mathrm{d}\mathbf{Q} \equiv \mathrm{d}x \wedge \mathrm{d}y \iff \frac{\partial \mathbf{P}}{\partial x}\frac{\partial \mathbf{Q}}{\partial w} - \frac{\partial \mathbf{P}}{\partial w}\frac{\partial \mathbf{Q}}{\partial x} \equiv \frac{\partial y}{\partial w} = 2w + 32x^{15}w^8 \\[8pt] \wedge\quad & \mathbf{P}(0,0) = \mathbf{Q}(0,0) = 0 \;\wedge\; \left( \frac{\partial \mathbf{P}}{\partial x}\frac{\partial \mathbf{Q}}{\partial y} - \frac{\partial \mathbf{P}}{\partial y}\frac{\partial \mathbf{Q}}{\partial x} \right)_{\mathrm{rel}} \equiv 1 \\[8pt] \wedge\quad & \det \begin{pmatrix} \frac{\partial}{\partial x}\left(\frac{\mathbf{P}+\mathbf{Q}}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{\mathbf{P}+\mathbf{Q}}{\sqrt{2}}\right) \\ \frac{\partial}{\partial x}\left(\frac{\mathbf{Q}-\mathbf{P}}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{\mathbf{Q}-\mathbf{P}}{\sqrt{2}}\right) \end{pmatrix}_{\mathrm{rel}} \equiv 1 \\[8pt] \wedge\quad & \mathbf{F}(0, 1) = \mathbf{F}(0, -1) = (0, 1) \implies \neg\operatorname{Inj}(\mathbf{F}) \\[8pt] \wedge\quad & [\mathbb{C}(\mathcal{S}) : \mathbb{C}(x, y)] = 9 \;\wedge\; [\mathbb{C}(\mathcal{S}) : \mathbb{C}(\mathbf{P}, \mathbf{Q})] = \deg(\mathbf{F}) = 32 \\[8pt] \wedge\quad & \mathcal{A}_{\mathbf{F}} = \{v = 0\} \;\wedge\; \mathcal{B}_{\mathbf{F}} = \{v = 0\} \cup \left\{ 256\left(\frac{16u}{15}\right)^{30}v^7 = 1 \right\} \\[8pt] \wedge\quad & \pi_1(\mathbb{C}^2 \setminus \mathcal{B}_{\mathbf{F}}) \curvearrowright \langle \sigma_0, \sigma_c \mid \sigma_0 = (2)(30), \, \sigma_c = (1\,2) \rangle \implies [\sigma_0, \sigma_c] \neq 1 \implies \text{Nem-abeli étale monodrómia} \\[8pt] \wedge\quad & \forall p \in \mathbb{P} : \quad \mathbf{P}^{(p)} = -w \sum_{k=0}^{p-2} (-xw)^k \implies \det J(\mathbf{P}^{(p)}, \mathbf{Q}) \equiv 1 \pmod p \quad \text{on } \mathbb{F}_p[x, w] \\[8pt] \wedge\quad & \widetilde{\mathbf{F}} \in \operatorname{Mor}\Big(\operatorname{Bl}_{\{[1:0:0], [0:1:0]\}}(\mathbb{P}^2) \setminus \widetilde{D}, \, \mathbb{P}^2\Big) \implies \frac{\mathrm{d}\mathbf{P} \wedge \mathrm{d}\mathbf{Q}}{\omega_{\mathrm{can}}} \equiv 1 \;\wedge\; \lim_{u \to 0} \mathbf{P}(u, t) = 0 \\[8pt] \wedge\quad & X_{\mathbf{P}} := \frac{\partial \mathbf{P}}{\partial x}\frac{\partial}{\partial y} - \frac{\partial \mathbf{P}}{\partial y}\frac{\partial}{\partial x} \implies \operatorname{depth}(\mathcal{T}_{\mathrm{BCW}}) = \infty \implies \neg\operatorname{LND}(X_{\mathbf{P}}) \\[8pt] \wedge\quad & \neg\operatorname{Proper}(\mathbf{F}) \;\wedge\; \neg\operatorname{Bir}(\mathbf{P}, \mathbf{Q}) \;\wedge\; \neg\operatorname{Inv}(\mathbf{P}, \mathbf{Q}) \end{aligned} $$