
vibhinn prakaar key sambandh

beejganitiya jyaamiti mein, ack affine beejganitiya samooh bahupad valay sootra _ 1 mein bahupadon key kuchh parivaar key beejganitiya roop seey bund kshetra parr samanya shoonya kaa samooh haye, ack affine vividhata yaa affine beejganitiya vividhata, ack affine beejganitiya samooh haye joe paribhaashit bahupadon dwara utpann aadarsh abhaajya haye.
kuchh granth kissy bhee beejganitiya samooh koo "vividhata" kahate hain, aur "aparivartaniya vividhata" ack beejganitiya samooh haye jiska paribhaashit aadarsh abhaajya haye (uparokt arth mein sambaddh vividhata).
kuchh sandarbhon mein (udaaharan key liye, hilbert kaa nalastelensaatz dekhein), yeh usa kshetra koo alag karane key liye upayogi haye jisamen gunank koo beejganitiya roop seey bund kshetra (yukt) seey maanaa jaataa haye, jiss parr samanya shoonya maane jaate hain (yaani, affine algebric sett key bindu hain). iss mamle mein, vividhata koo "paribhaashit kiya gayaa" kahaa jaataa haye, aur vividhata key joe bindu sambandhit hain unhein "-tarksangat" yaa "tarksangat" kahaa jaataa haye. samanya mamle mein jahan vaastavik sankhyaon kaa kshetra haye, ack-tarksangat bindu koo "vaastavik bindu" kahaa jaataa haye. jaba kshetra nirdisht naheen hotaa haye, tou ack "parimeya bindu" ack aisaa bindu hotaa haye joe parimeya sankhyaon parr parimeya hotaa haye. udaaharan key liye, firmate kii antim pramey kaa kahana haye kii dou seey adhik purnank key liye koee tarksangat bindu naheen haye.
parichay.
ack affine beejganitiya samooh bahupad samikaranon kii pranali key beejganitiya roop seey bund kshetra mein samadhanon kaa samooh haye jisamen gunank hotey hain. adhik sateek roop seey, yadi sootra _ 2 gunank key saath bahupad hain, tou way ack affine beejganitiya sett koo paribhaashit karate hain. 
ack affine (beejganitiya) vividhata ack affine beejganitiya samooh haye joe dou uchit affine beejganitiya upasamuchchay kaa sangh naheen haye. iss tarah key ack affine beejganitiya samooh koo aksar "irreducible" kahaa jaataa haye. 
yadi ack affine beejganitiya samooh haye, aur sabhi bahupadon kaa aadarsh haye joe shoonya parr hain, tou bhagfal valay sootra _ 4 koo "x" kaa sootra kahaa jaataa haye. yadi "x" ack affine vividhata haye, tou "i" prime haye, isaliye nirdeshank ring ack abhinn domen haye. nirdeshank valay "r" key tatvon koo vividhata parr "niyamit kaary" yaa "bahupad kaary" bhee kahaa jaataa haye. way vividhata parr "niyamit karyon kii valay" banaate hain, yaa, buss, "vividhata kii valay"; dusare shabdon mein (dekhein #structure kosh), yeh "x" kii sanrachana kosh key vaishvik khandon kaa sthaan haye.
ack kism kaa ayam pratyek kism seey judaa ack purnank haye, aur yahaan taka kii pratyek beejganitiya samooh seey bhee, jiska mahatva isaki samatulya paribhashaon kii badee sankhya parr nirbhar kartaa haye (ack beejganitiya vividhata kaa ayam dekhein).
tarksangat bindu.
beejganitiya roop seey bund kshetra parr ack sambaddh vividhata sootra key liye, aur ack upakshetra key liye, ack-"tarksangat bindu" ack bindu sootra haye joe kii 10 haye, jiska ack bindu jisake nirdeshank key tatva hain. ack sambaddh kism key-tarksangat binduon key sangrah koo aksar sootra 11 seey darshaaya jaataa haye, yadi aadhar kshetra jatil sankhya haye, tou joe bindu-tarksangat hain (jahaan vaastavik sankhya haye) unhein vividhata key "vaastavik bindu" kahaa jaataa haye, aur-tarksangat binduon (tarksangat sankhya) koo aksar kewal "tarksangat bindu" kahaa jaataa haye.
udaaharan key liye, yeh vibhinn sootra _ 12 kaa ack-tarksangat aur ack-tarksangat bindu haye kyonki yeh purnank mein haye aur isake sabhi nirdeshank purnank hain. yeh ack vaastavik bindu haye joe-tarksangat naheen haye, aur sootra _ 13 ack bindu haye joe-tarksangat naheen haye. iss kism koo vritt kahaa jaataa haye, kyonki isake-tarksangat binduon kaa samooh ikai vritt haye. isamen aseem roop seey kaee tarksangat bindu hain joe ank hain. 
jahan ack parimeya sankhya haye.
vritt sootra _ 15 degree dou key beejganitiya vakr kaa ack udaaharan haye jiska koee-tarksangat bindu naheen haye. yeh iss tathya seey anuman lagaaya jaa sakata haye kii, modulo, dou vargon kaa yog naheen hoe sakata haye.
yeh saabit kiya jaa sakata haye kii ack-tarksangat bindu key saath degree dou key beejganitiya vakr mein anant roop seey kaee anya-tarksangat bindu hotey hain; aisaa pratyek bindu vakr kaa doosara pratichhedan bindu haye aur ack tarksangat dhalaan key saath ack rekha joe tarksangat bindu seey gujarati haye.
jatil vividhata sootra _ 16 mein koee tarksangat bindu naheen hain, lekin kaee jatil bindu hain.
yadi jatil sankhyaon parr paribhaashit ack vishisht vividhata haye, tou isake-tarksangat binduon koo kagaj key tukade parr yaa graphing software dwara kheencha jaa sakata haye. dahini or kaa chitra sootra _ 17 key-tarksangat binduon koo darshaata haye.
ekvachan bindu aur sparshrekha sthaan.
bahupad sootra _ 18 dwara paribhaashit ack affine vividhata aur sootra _ 19 kaa ack bindu honah chaahiye.
aitt kaa jacobian matricks aanshik vyutpannon kaa matricks haye. 
yadi crum kaa crum, isake samhitaakaran key barabar haye tou bindu "niyamit" haye, aur anyatha "ekvachan".
yadi niyamit haye, tou sparshrekha sthaan sootra _ 8 kaa sambaddh upasthaan haye jise raikhik samikaranon dwara paribhaashit kiya gayaa haye.
yadi bindu ekvachan haye, tou inn samikaranon dwara paribhaashit affine upasthaan koo kuchh lekhakon dwara sparshrekha sthaan bhee kahaa jaataa haye, jabaki anya lekhakon kaa kahana haye kii ekvachan bindu parr koee sparshrekha sthaan naheen haye.
ack adhik aantarik paribhasha, joe nirdeshank kaa upayog naheen karati haye, zariski sparshrekha sthaan dwara dee jaati haye.
zariski topology.
key" aine "key affine beejganitiya samooh" key "aine" parr ack topology key bund sett banaate hain, jinhen zariski topology kahaa jaataa haye. yeh iss tathya seey anusaran kartaa haye kii sootra _ 23 sootra _ 24 sootra _ 25 aur sootra _ 26 (vaastav mein, affine beejganitiya samuchchay kaa ack ganana yogya pratichhedan ack affine beejganitiya samuchchay haye).
zariski topology koo buniyaadi khule seton key madhyam seey bhee varnit kiya jaa sakata haye, jahaan zariski-open sett formula _ 27 key seton key ganana yogya sangh hain, formula _ 28 key liye yeye buniyaadi khule sett ack ekal bahupad key bund sett sootra _ 29 shoonya sthaan key "k" n "mein poorak hain. yadi "k" notherian haye (udaaharan key liye, yadi "k" ack kshetra yaa ack pramukh aadarsh kshetra haye), tou "k" kaa pratyek aadarsh parimit roop seey utpann hotaa haye, isaliye pratyek khula sett buniyaadi khule seton kaa ack seemit sangh haye.
yadi "v" "k" "n" kii ack upaparivartan haye tou "v" parr zariski topology kewal "k" "n" parr zariski topology seey viraasat mein millie upasthaan topology haye.
jyaamiti-beejganit patrachar.
ack affine kism kii jyaamitiya sanrachana isake nirdeshank valay kii beejganitiya sanrachana seey gahare tareeke seey judee hui haye. i" aur "j" koo "k [v]" key aadarsh honey dein, joe ack affine kism "v" kaa nirdeshank valay haye. maan leejiye kii "i (v)" sootra _ 30 mein sabhi bahupadon kaa samooh haye joe "v" parr gayab hoe jaataa haye, aur sootra _ 31 aadarsh "i" key moolank koo darshaata haye, bahupadon kaa samooh "f" jisake liye "f" kii kuchh shakti "i" mein haye. aadhar kshetra koo beejganitiya roop seey bund karane kaa kaaran yeh haye kii affine kismein swachaalit roop seey hilbert key nalastelensaatz koo santusht karati hainh sootra _ 30 mein ack aadarsh "jey" key liye jahaan "key" ack beejganitiya roop seey bund kshetra haye, sootra _ 33
k [v]" key mool aadarsh (joe apane mool hain) "v" key beejganitiya upasamuchchay key anuroop hain. vaastav mein, mool aadarshon "i" aur "j" key liye, sootra 34 yadi aur kewal yadi sootra _ 35 isaliye "v (i) = v (j)" yadi aur kewal yadi "i = j". isake alawa, ack sambaddh beejganitiya samooh "doubleu" lene aur "aaee (doubleu)" vaapas karane vaalaa kaary, sabhi karyon kaa samooh joe "doubleu" key sabhi binduon parr bhee gayab hoe jaataa haye, ack mool siddhant koo ack beejganitiya sett koo nirdisht karane vaale kaary kaa vyutkram haye, nalstellensatz dwara. isaliye affine beejganitiya samuchchay aur mool aadarshon key beach kaa patrachar ack dwibhajan haye. ack affine beejganitiya samooh kaa nirdeshank valay cumm hoe jaataa haye (neelpotent-mukt), kyonki ack aadarsh "i" ack valay "r" mein mool haye yadi aur kewal tabhi jaba bhagfal valay "r/i" cumm hoe jaae.
nirdeshank valay key pramukh aadarsh upa-prakaron seey mell khaate hain. ack sambaddh beejganitiya samooh "v (i)" koo dou anya beejganitiya seton key sangh key roop mein likhaa jaa sakata haye yadi aur kewal tabhi jaba uchit aadarshon "j" key liye "i = jk" aur "k" "i" key barabar naa hoe (jiss mamle mein sootra _ 36). yeh mamla tabhi haye jaba "i" abhaajya naheen haye. affine upa-prakaar way hain jinka nirdeshank valay ack abhinn kshetra haye. aisaa isaliye haye kyonki ack aadarsh taba abhaajya hotaa haye jaba aadarsh dwara valay kaa bhagfal ack abhinn kshetra hoe.
k [v]" key adhikatam aadarsh "v" key binduon key anuroop hain. yadi "i" aur "j" mool aadarsh hain, tou sootra 37 yadi aur kewal yadi sootra 38 adhikatam aadarsh key roop mein mool hain, tou adhikatam aadarsh nyuntam beejganitiya samuchchay (jinamen koee uchit beejganitiya upasamuchchay naheen haye) key anuroop hain, joe "v" mein bindu hain. yadi "v" nirdeshank valay sootra key saath ack sambaddh vividhata haye tou yeh patrachar maanchitra sootra 40 key madhyam seey spasht hoe jaataa haye jahaan sootra _ 41 bahupad sootra _ 42 key bhagfal beejganit "r" mein chhavi koo darshaata haye ack beejganitiya subset ack bindu haye yadi aur kewal tabhi jaba upasamuchchay kaa nirdeshank valay ack kshetra haye, kyonki ack adhikatam aadarsh dwara ack valay kaa bhagfal ack kshetra haye.
nimnalikhit talika iss patrachar koo ack sambaddh vividhata key beejganitiya upasamuchchay aur sambandhit nirdeshank valay key aadarshon key liye sankshep mein bataati haih
affine kismon key utpaad.
samroopata kaa upayog karke samroopata kii kismon key utpaad koo paribhaashit kiya jaa sakata haye aur phir iss nae samroopata sthaan mein utpaad koo embed kiya jaa sakata haye. maan leejiye aur kramshah nirdeshank valay houn, taaki unake utpaad mein samanvaya valay hoe. maan leejiye kii yeh ack beejganitiya upasamuchchay haye aur ack beejganitiya upasamuchchay haye tou pratyek ack bahupad haye, aur pratyek mein haye. utpaad mein beejganitiya samooh key roop mein paribhaashit kiya gayaa haye aur yadi pratyek, aparivartaniya haye tou aparivartaniya haye.
zariski topology donon sthaanon parr zariski topology kaa topological utpaad naheen haye. vaastav mein, utpaad topology buniyaadi khule seton key utpaadon dwara utpann kii jaati haye aur isaliye, bahupad joe ack bahupad key saath ack bahupad key utpaad key roop mein praapt naheen kiye jaa sakate hain, way beejganitiya seton koo paribhaashit karenge joe zariski topology mein hain lekin utpaad topology mein naheen hain.
affine kismon key rupankan.
ack aakritivaad, yaa niyamit maanchitra, affine kismon kaa ack kaary haye joe pratyek nirdeshank mein bahupad haih adhik sateek roop seey, affine kismon key liye aur, seey ack aakritivaad usa roop kaa ack maanchitra haye jahan pratyek key liye yeye affine kismon kii shreni mein aakriti hain.
beejganitiya roop seey bund kshetra parr affine kismon key rupankan aur vipareet disha mein jaane vaale affine kismon key nirdeshank valayon key samroopata key beach ack-seey-ack patrachar haye. isake kaaran, iss tathya key saath kii affine kismon aur unake samanvaya valay key beach ack-seey-ack patrachar haye, affine kismon kii shreni affine kismon key samanvaya valay kii shreni key liye dohri haye, joe kii affine kismon key samanvaya valay kii shreni parr haye, joe sateek roop seey parimit roop seey utpann, neelpotent-mukt algebra kii shreni haye. 
adhik sateek roop seey, affine kismon key pratyek rupankan key liye, nirdeshank valay (vipareet disha mein jaane vaale) key beach ack samroopata haye, aur iss tarah key pratyek samroopata key liye, nirdeshank valay seey judee kismon kaa ack rupankan haye. ise spasht roop seey dikhaaya jaa sakata haih chalo aur kramshah nirdeshank valay key saath prakaron koo sanlagn karein. ack roopvaad honey dein. vaastav mein, bahupad ring karakon key beach ack samroopata, joe ring key madhyam seey vishisht roop seey hoti haye aur ack samroopata, pratyek samroopata key chitron dwara vishisht roop seey nirdhaarit kii jaati haye, pratyek key liye chhavi key chayan key liye vishisht roop seey mell khaati haye. phir kissy bhee rupankan koo ack samroopata seey diyaa jaa sakata haye joe sootra _ 43 koo bhejata haye jahan sootra _ 44 mein samatulya varg haye. 
isi tarah, nirdeshank valay key pratyek samroopata key liye, affine kismon key ack rupankan kaa nirmaan vipareet disha mein kiya jaa sakata haye. uparokt anuchhed koo pratibimbit karate hue, ack samroopata ack bahupad sootra _ 45 inch koo bhejati haye. yeh dwara paribhaashit kismon key rupankan seey mell khaata haye 
sanrachana sheef.
neeche varnit sanrachana sheef seey susajjit, ack affine kism sthaaniya roop seey ghoornit sthaan haye.
ack affine kism "x" koo nirdeshank valay "a" key saath diyaa gayaa haye, "k"-algebra sootra _ 46 key sheef koo sootra _ 47 koo "u" parr niyamit karyon kii valay honey dekar paribhaashit kiya gayaa haye.
maan leejiye "d" ("f") = {"x" . "f" ("" x "") ≤0} pratyek "" f "" key liye "" a "" mein.  way "x" key topology key liye ack aadhar banaate hain aur isaliye sootra _ 46 khule sett "d" ("f") parr isake moolyon dwara nirdhaarit kiya jaataa haye. (yeh bhee dekhein kii module seey judaa modules#sheaf kaa sheef.)
mukhya tathya, joe aavashyak tareeke seey hilbert nalstellensatz parr nirbhar kartaa haye, nimnalikhit haih
pramanah samavesh spasht haye. isake vipareet, "g" koo baain or aur sootra _ 49 mein honey dein, joe ack aadarsh haye. yadi "x" "d" ("f") mein haye, tou, choonki "g" "" x "key paas niyamit haye, isaliye" x "kaa kuchh khula pados" d "(" h ") haye jaise kii sootra _ 50; arthaat," h "" m "" g "" a "mein haye aur iss prakaar" x "" v "(" j ") mein naheen haye. dusare shabdon mein, sootra _ 51 aur iss prakaar hilbert nalstellensatz kaa tatparya haye kii "f" "j" key moolank mein haye; i.ee.sootra _ 52. sootra _ 53
daavaa, sabse pahale, kaa tatparya haye kii "x" ack "sthaaniya roop seey valayabaddh" sthaan haye kyonki
jahan sootra _ 55. doosara, daave kaa tatparya haye kii sootra _ 46 ack sheef haye; vaastav mein, yeh kahata haye kii yadi koee kaary "d" ("f") parr niyamit (bindu kii or) haye, tou yeh "d" ("f") key nirdeshank valay mein honah chaahiye; yaani, "niyamitata" koo ack saath patch kiya jaa sakata haye.
isaliye, sootra _ 57 ack sthaaniya roop seey valayabaddh sthaan haye.
aatmiyata parr sair kii pramey.
sair kii ack pramey ack affine vividhata kaa ack samarupatapurna lakshan varnan detee haye; yeh kahata haye kii ack beejganitiya vividhata kewal tabhi prabhaavit hoti haye jaba kissy sootra key liye sootra _ 58 aur "x" parr koee ardh-susangat sheef "f" hoe. (see. aff.) carton kii pramey bee.) yeh ack saha-vaigyaanik adhyayan koo ack saha-vaigyaanik vividhata key gair-astitva mein banaata haye, joe kii projective mamle key vipareet haye jisamen line bandalon key saha-vigyaan samooh kendriya ruchi key hotey hain.
beejganitiya samoohon koo jodein.
beejganitiya roop seey bund kshetra parr ack affine vividhata koo ack affine beejganitiya samooh kahaa jaataa haye yadi isamen haih
ack saath, yeye vividhata parr ack samooh sanrachana koo paribhaashit karate hain. uparokt rupankan koo aksar sadharan samooh sanketan kaa upayog karke likhaa jaataa haih ise iss prakaar likhaa jaa sakata haye, yaa; vyutkram koo gunak sanketan key roop mein yaa iska upayog karke likhaa jaa sakata haye, sahakriyashilta, pehchaan aur vyutkram niyamon koo iss prakaar phir seey likhaa jaa sakata haih, aur.
ack affine beejganitiya samooh kaa sabse pramukh udaaharan degree kaa samanya raikhik samooh haye yeh vector space key raikhik parivartanon kaa samooh haye yadi iska ack aadhar nishchit haye, tou yeh pravishtiyon key saath vyutkramaniya matricks key samooh key barabar haye, yeh dikhaaya jaa sakata haye kii koee bhee affine beejganitiya samooh ack upsamooh key liye samroop haye. iss kaaran seey, sambaddh beejganitiya samoohon koo aksar raikhik beejganitiya samooh kahaa jaataa haye. 
affine beejganitiya samooh parimit saral samoohon key vargeekaran mein ack mahatvapurna bhoomika nibhaate hain, kyonki jhooth prakaar key samooh ack affine beejganitiya samooh key sabhi samkon-tarksangat bindu hotey hain, jahaan ack seemit kshetra hotaa haye.
sandarbh.
mool lekh sambandhit francisi lekh key aanshik maanav anuvaad key roop mein likhaa gayaa thaa.