Lemma 27.13.6 (Segre embedding). Let $S$ be a scheme. There exists a closed immersion

called the *Segre embedding*.

Lemma 27.13.6 (Segre embedding). Let $S$ be a scheme. There exists a closed immersion

\[ \mathbf{P}^ n_ S \times _ S \mathbf{P}^ m_ S \longrightarrow \mathbf{P}^{nm + n + m}_ S \]

called the *Segre embedding*.

**Proof.**
It suffices to prove this when $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Hence we will drop the index $S$ and work in the absolute setting. Write $\mathbf{P}^ n = \text{Proj}(\mathbf{Z}[X_0, \ldots , X_ n])$, $\mathbf{P}^ m = \text{Proj}(\mathbf{Z}[Y_0, \ldots , Y_ m])$, and $\mathbf{P}^{nm + n + m} = \text{Proj}(\mathbf{Z}[Z_0, \ldots , Z_{nm + n + m}])$. In order to map into $\mathbf{P}^{nm + n + m}$ we have to write down an invertible sheaf $\mathcal{L}$ on the left hand side and $(n + 1)(m + 1)$ sections $s_ i$ which generate it. See Lemma 27.13.1. The invertible sheaf we take is

\[ \mathcal{L} = \text{pr}_1^*\mathcal{O}_{\mathbf{P}^ n}(1) \otimes \text{pr}_2^*\mathcal{O}_{\mathbf{P}^ m}(1) \]

The sections we take are

\[ s_0 = X_0Y_0, \ s_1 = X_1Y_0, \ldots , \ s_ n = X_ nY_0, \ s_{n + 1} = X_0Y_1, \ldots , \ s_{nm + n + m} = X_ nY_ m. \]

These generate $\mathcal{L}$ since the sections $X_ i$ generate $\mathcal{O}_{\mathbf{P}^ n}(1)$ and the sections $Y_ j$ generate $\mathcal{O}_{\mathbf{P}^ m}(1)$. The induced morphism $\varphi $ has the property that

\[ \varphi ^{-1}(D_{+}(Z_{i + (n + 1)j})) = D_{+}(X_ i) \times D_{+}(Y_ j). \]

Hence it is an affine morphism. The corresponding ring map in case $(i, j) = (0, 0)$ is the map

\[ \mathbf{Z}[Z_1/Z_0, \ldots , Z_{nm + n + m}/Z_0] \longrightarrow \mathbf{Z}[X_1/X_0, \ldots , X_ n/X_0, Y_1/Y_0, \ldots , Y_ n/Y_0] \]

which maps $Z_ i/Z_0$ to the element $X_ i/X_0$ for $i \leq n$ and the element $Z_{(n + 1)j}/Z_0$ to the element $Y_ j/Y_0$. Hence it is surjective. A similar argument works for the other affine open subsets. Hence the morphism $\varphi $ is a closed immersion (see Schemes, Lemma 26.4.2 and Example 26.8.1.) $\square$

Your email address will not be published.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.

## Comments (2)

Comment #5998 by Nick on

Comment #6163 by Johan on