The Stacks project

Lemma 33.35.2. Let $k$ be a field. Let $n \geq 1$. Let $i : H \to \mathbf{P}^ n_ k$ be a hyperplane. Then there exists an isomorphism

\[ \varphi : \mathbf{P}^{n - 1}_ k \longrightarrow H \]

such that $i^*\mathcal{O}(1)$ pulls back to $\mathcal{O}(1)$.

Proof. We have $\mathbf{P}^ n_ k = \text{Proj}(k[T_0, \ldots , T_ n])$. The section $s$ corresponds to a homogeneous form in $T_0, \ldots , T_ n$ of degree $1$, see Cohomology of Schemes, Section 30.8. Say $s = \sum a_ i T_ i$. Constructions, Lemma 27.13.7 gives that $H = \text{Proj}(k[T_0, \ldots , T_ n]/I)$ for the graded ideal $I$ defined by setting $I_ d$ equal to the kernel of the map $\Gamma (\mathbf{P}^ n_ k, \mathcal{O}(d)) \to \Gamma (H, i^*\mathcal{O}(d))$. By our construction of $Z(s)$ in Divisors, Definition 31.14.8 we see that on $D_{+}(T_ j)$ the ideal of $H$ is generated by $\sum a_ i T_ i/T_ j$ in the polynomial ring $k[T_0/T_ j, \ldots , T_ n/T_ j]$. Thus it is clear that $I$ is the ideal generated by $\sum a_ i T_ i$. Note that

\[ k[T_0, \ldots , T_ n]/I = k[T_0, \ldots , T_ n]/(\sum a_ i T_ i) \cong k[S_0, \ldots , S_{n - 1}] \]

as graded rings. For example, if $a_ n \not= 0$, then mapping $S_ i$ equal to the class of $T_ i$ works. We obtain the desired isomorphism by functoriality of $\text{Proj}$. Equality of twists of structure sheaves follows for example from Constructions, Lemma 27.11.5. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 33.35: Coherent sheaves on projective space

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Your email address will not be published.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 089Z. Beware of the difference between the letter 'O' and the digit '0'.