View source for Segre imbedding

The imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838001.png" /> of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838002.png" /> of projective spaces into the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838003.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838004.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838006.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838007.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838008.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838009.png" />) are homogeneous coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380010.png" />, then the mapping is defined by the formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380012.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380013.png" /> is well-defined and is a closed imbedding. The image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380014.png" /> of a Segre imbedding is called a Segre variety. The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380015.png" /> has a simple geometrical meaning: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380016.png" /> is the non-singular quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380017.png" /> with equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380018.png" />. The images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380020.png" /> give two families of generating lines of the quadric.

The terminology is in honour of B. Segre.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>