Exercise 111.51.4. Consider the projective1 variety

${\mathbf P}^1 \times {\mathbf P}^1 = {\mathbf P}^1_{{\mathbf C}} \times _{\mathop{\mathrm{Spec}}({\mathbf C})} {\mathbf P}^1_{\mathbf C}$

over the field of complex numbers ${\mathbf C}$. It is covered by four affine pieces, corresponding to pairs of standard affine pieces of ${\mathbf P}^1_{\mathbf C}$. For example, suppose we use homogeneous coordinates $X_0, X_1$ on the first factor and $Y_0, Y_1$ on the second. Set $x = X_1/X_0$, and $y = Y_1/Y_0$. Then the 4 affine open pieces are the spectra of the rings

${\mathbf C}[x, y], \quad {\mathbf C}[x^{-1}, y], \quad {\mathbf C}[x, y^{-1}], \quad {\mathbf C}[x^{-1}, y^{-1}].$

Let $X \subset {\mathbf P}^1 \times {\mathbf P}^1$ be the closed subscheme which is the closure of the closed subset of the first affine piece given by the equation

$y^3(x^4 + 1) = x^4 -1.$

1. Show that $X$ is contained in the union of the first and the last of the 4 affine open pieces.

2. Show that $X$ is a nonsingular projective curve.

3. Consider the morphism $pr_2 : X \to {\mathbf P}^1$ (projection onto the first factor). On the first affine piece it is the map $(x, y) \mapsto x$. Briefly explain why it has degree $3$.

4. Compute the ramification points and ramification indices for the map $pr_2 : X \to {\mathbf P}^1$.

5. Compute the genus of $X$.

[1] The projective embedding is $((X_0, X_1), (Y_0, Y_1))\mapsto (X_0Y_0, X_0Y_1, X_1Y_0, X_1Y_1)$ in other words $(x, y)\mapsto (1, y, x, xy)$.

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