# Segre map

The *Segre map* is an embedding $s:{\mathbb{P}}^{n}\times {\mathbb{P}}^{m}\to {\mathbb{P}}^{nm+n+m}$ of the product of
two projective spaces^{} into a larger projective space. It is important since it makes the
product of two projective varieties into a projective variety again.
Invariantly, it can described as follows. Let $V,W$ be (finite dimensional) vector spaces; then

$$\begin{array}{cccccc}\hfill s:& \hfill \mathbb{P}V& \hfill \times \hfill & \mathbb{P}W\hfill & \hfill \u27f6\hfill & \hfill \mathbb{P}(V\otimes W)\hfill \\ & \hfill [x]& \hfill ,\hfill & [y]\hfill & \hfill \u27fc\hfill & \hfill [x\otimes y]\hfill \end{array}$$ |

In homogeneous coordinates, the pair of points $[{x}_{0}:{x}_{1}:\mathrm{\cdots}:{x}_{n}]$, $[{y}_{0}:{y}_{1}:\mathrm{\cdots}:{y}_{m}]$ maps to

$$[{x}_{0}{y}_{0}:{x}_{1}{y}_{0}:\mathrm{\cdots}:{x}_{n}{y}_{0}:{x}_{0}{y}_{1}:{x}_{1}{y}_{1}:\mathrm{\cdots}:{x}_{n}{y}_{m}].$$ |

If we imagine the target space as the projectivized version of the space of $(n+1)\times (m+1)$ matrices, then the image is exactly the set of matrices which have rank 1; thus it is the common zero locus of the equations

$$\left|\begin{array}{cc}\hfill {a}_{ij}\hfill & \hfill {a}_{il}\hfill \\ \hfill {a}_{kj}\hfill & \hfill {a}_{kl}\hfill \end{array}\right|={a}_{ij}{a}_{kl}-{a}_{il}{a}_{kj}=0$$ |

for all $$, $$. Varieties^{} of this form (defined by vanishing
of minors in some space of matrices) are usually called *determinantal varieties*.

Title | Segre map |
---|---|

Canonical name | SegreMap |

Date of creation | 2013-03-22 14:24:45 |

Last modified on | 2013-03-22 14:24:45 |

Owner | halu (5781) |

Last modified by | halu (5781) |

Numerical id | 4 |

Author | halu (5781) |

Entry type | Definition |

Classification | msc 14A25 |

Classification | msc 14M12 |

Synonym | Segre embedding |