# 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

 $\begin{array}[]{rrclcc}s:&\mathbb{P}V&\times&\mathbb{P}W&\longrightarrow&% \mathbb{P}(V\otimes W)\\ &[x]&,&[y]&\longmapsto&[x\otimes y]\end{array}$

In homogeneous coordinates, the pair of points $[x_{0}:x_{1}:\cdots:x_{n}]$, $[y_{0}:y_{1}:\cdots:y_{m}]$ maps to

 $[x_{0}y_{0}:x_{1}y_{0}:\cdots:x_{n}y_{0}:x_{0}y_{1}:x_{1}y_{1}:\cdots:x_{n}y_{% m}].$

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

 $\left|\begin{array}[]{cc}a_{ij}&a_{il}\\ a_{kj}&a_{kl}\end{array}\right|=a_{ij}a_{kl}-a_{il}a_{kj}=0$

for all $0\leq i, $0\leq j. Varieties of this form (defined by vanishing of minors in some space of matrices) are usually called determinantal varieties.

Title Segre map SegreMap 2013-03-22 14:24:45 2013-03-22 14:24:45 halu (5781) halu (5781) 4 halu (5781) Definition msc 14A25 msc 14M12 Segre embedding
{"ft_lang_label":"__label__en","ft_lang_prob":0.85387075,"math_prob":0.9995913,"size":1016,"snap":"2020-10-2020-16","text_gpt3_token_len":264,"char_repetition_ratio":0.12055336,"word_repetition_ratio":0.012578616,"special_character_ratio":0.26181102,"punctuation_ratio":0.08695652,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998387,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-10T06:18:58Z\",\"WARC-Record-ID\":\"<urn:uuid:e635d2c3-1d7f-4997-a974-3fe5092a8ede>\",\"Content-Length\":\"13383\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:228fc807-e121-4507-9535-03fe5278a870>\",\"WARC-Concurrent-To\":\"<urn:uuid:76d0e4f3-d8f2-4651-b1b4-cf807cd2ffc7>\",\"WARC-IP-Address\":\"129.97.206.129\",\"WARC-Target-URI\":\"https://planetmath.org/SegreMap\",\"WARC-Payload-Digest\":\"sha1:46I44IBQ75UPHUTPHXJGNZQNDFHOLGVS\",\"WARC-Block-Digest\":\"sha1:LII4YE46FW7THMV76JAAYSPMZSMUVXQU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371886991.92_warc_CC-MAIN-20200410043735-20200410074235-00429.warc.gz\"}"}