The Stacks project

Exercise 111.34.3. Let $A, B, C \in \mathbf{C}[t]$ be nonzero polynomials. Consider the morphism of schemes

\[ \pi : X = \mathop{\mathrm{Spec}}(\mathbf{C}[x, y, t]/(A + Bx^2 + Cy^2)) \longrightarrow S = \mathop{\mathrm{Spec}}(\mathbf{C}[t]). \]

Show there does exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$. (Hint: Symbolically, write $x = X/Z$, $y = Y/Z$ for some $X, Y, Z \in \mathbf{C}[t]$ of degree $\leq d$ for some $d$, and work out the condition that this solves the equation. Then show, using dimension theory, that if $d >> 0$ you can find nonzero $X, Y, Z$ solving the equation.)


Comments (0)

There are also:

  • 2 comment(s) on Section 111.34: Morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0296. Beware of the difference between the letter 'O' and the digit '0'.