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Tag 02FJ

Chapter 102: Exercises > Section 102.48: Schemes, Final Exam, Fall 2007

Exercise 102.48.4. 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 $(x, y)\mapsto (1, y, x, xy)$.

The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 4860–4900 (see updates for more information).

\begin{exercise}
\label{exercise-curve-p1-p1}
Consider the
projective\footnote{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 $(x, y)\mapsto (1, y, x, xy)$.} variety
$$
{\mathbf P}^1 \times {\mathbf P}^1 = {\mathbf P}^1_{{\mathbf C}}
\times_{\Spec({\mathbf C})} {\mathbf P}^1_{\mathbf C}
$$
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
$$
y^3(x^4 + 1) = x^4 -1.
$$
\begin{enumerate}
\item Show that $X$ is contained in the union of the first and
the last of the 4 affine open pieces.
\item Show that $X$ is a nonsingular projective curve.
\item 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 $3$.
\item Compute the ramification points and ramification indices
for the map $pr_2 : X \to {\mathbf P}^1$.
\item Compute the genus of $X$.
\end{enumerate}
\end{exercise}

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