\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

Exercise 103.50.4. 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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