
cubic uparitalam

ganita saastramlo, cubic uparitalam anedi 3-dimensional space loe unna uparitalam, idhi degree 3 yokka oka bahupadi sameekaranam dwara nirvachinchabadutundi. cubic uparitalaalu beejaganita jyaamitiloo praathamika udaaharanalu. eee siddhaantam anubandha sthalamlo kakunda prakshepaka sthalamlo panicheyadam dwara saralikrutam cheyabadindi, kabatti cubic uparitalaalu saadhaaranamgaa prakshepaka 3-antariksha suutramloe pariganinchabadataayi. siddhaantam kooda vaastava sankhyala kante sanklishta sankhyalapai uparitalaalapai drushti pettadam dwara marinta ekareetigaa maarutundi; sanklishta uparitalaaniki vaastava parimaanam 4 undani gamaninchandi. oka saadhaarana udaaharana fermat cubic uparitalam.
suutram _ 1 loe. ghana uparitalaala yokka aneka lakshanaalu saadhaaranamgaa dell pejo uparitalaala kosam untaayi.
cubic uparitalaala hetubaddhata.
beejaganitaparamgaa moosivesina kshetrampai mruduvaina ghana uparitalaalu "x" yokka pradhana lakshanam emitante, 1866loe alfred clebs chuupina vidhamgaa avanni hetubaddhamainavi. antey, projective plain formula _ 4 mainus takkuva-dimensional upasamiti mariyu "x" mainus takkuva-dimensional upasamiti madhya hetubaddhamaina vidhula dwara nirvachinchabadina okariki okaru sambandham undhi. marinta saadhaaranamgaa, beejaganitaparamgaa moosivesina kshetrampai prati irreducible cubic uparitalam (bahusha ekavachanam) cubic vakrarekhapai projective cone ayithe tappa hetubaddhamainadi. eee vishayamlo, cubic uparitalaalu suutram _ 1 loe kaneesam 4 degreela mruduvaina uparitalaala kante chaalaa saralamgaa untaayi, ivi eppuduu hetubaddhamainavi kaavu. lakshana sunnaalo, suutram _ 1loe kaneesam 4 degreela mruduvaina uparitalaalu kooda niyantrinchabadavu.
marinta balamgaa, beejaganitaparamgaa moosivesina kshetrampai suutram _ 1 looni prati mruduvaina ghana uparitalam 6 paayintla vadda suutram _ 4 yokka paeluduku isomorphic anie clebs choopinchaadu. phalitamgaa, sanklishta sankhyalapai prati mruduvaina ghana uparitalam anusandhaaninchabadina mottam suutram _ 9kii bhinnamainadi, ikkada mainus gurthu dhorani maarpunu suuchistumdi. dheeniki viruddhamgaa, 6 paayintla vadda suutram _ 4 yokka paeludu anedi cubic uparitalaaniki isomorphicga untundi, okavela paayintlu saadhaarana sthithilo unte, antey moodu paayintlu oka rekhapai undavu mariyu mottam 6 oka sankuvu meeda undavu. sanklishtamaina aneka retlu (ledhaa beejaganita vaividhyam) gaa, uparitalam aaa 6 binduvula amarikapai aadharapadi untundi.
cubic uparitalampai 27 panktulu.
cubic uparitalaala hetubaddhataku chaalaa rujuvulu uparitalampai oka geetanu kanugonadam dwara praarambhamavutaayi. (projective jyaamiti sandarbhamlo, suutram _ 1 looni oka rekha suutram _ 12kii isomorphic.) marinta khachitamgaa, orther kylie mariyu georg salmon 1849loe beejaganitamgaa moosivesina kshetrampai prati mruduvaina cubic uparitalam sarigga 27 panktulanu kaligi undani chuupimchaaru. idhi cubics yokka vilakshanhamaina lakshanamh mruduvaina quadrick (degree 2) uparitalam nirantara rekhala kutumbamtho kappabadi untundi, ayithe formula _ 1 loe kaneesam 4 degreela uparitalaalu etuvanti rekhalanu kaligi undavu. 27 panktulanu kanugonadaaniki maroka upayogakaramaina saanketikata shubert calculus, idhi suutram _ 1 paie unna rekhala gaddi yokka khandana siddhaantaanni upayoginchi rekhala sankhyanu lekkistundi.
mruduvaina sanklishta cubic uparitalam yokka gunakaalu maarutuu untaayi kabatti, 27 panktulu nirantaram kadulutaayi. phalitamgaa, mruduvaina cubic uparitalaala kutumbamlo oka closed loop 27 rekhala kramaparivartananu nirnayistundi. eee vidhamgaa utpannamayye 27 rekhala kramaparivartana samuuhaanni cubic uparitalaala kutumbaaniki chendina monodromy samuuham antaaru. 19va sataabdapu oka visaeshamaina aavishkarana emitante, monodromy samuuham chinnadi kaadu ledhaa mottam cemetric group formula kaadu; idhi 51840 kramamlo unna samuuham, idhi rekhala samitipai parivartanagaa panichestundi. eee samuuhaanni kramamga gurtinchaaru (eli carton (1896), orther cobul (1915-17), mariyu patrick du wall (1936))), idhi vail group off type formula _ 16 gaa gurtinchabadindi, idhi 6-dimensional real vector space paie pratibimbaala dwara utpatti cheyabadina samuuham, idhi dimension 78 yokka lye group formula _ 16 kii sambandhinchinadi.
51840 kramamlo unna oke samuuhaanni 27 panktula graph yokka automorphism samuuhamgaa, prati rekhaku oka sheersham mariyu rendu panktulu kalisinappudalla oka anchutho, kalayika paramgaa varninchavachhu. eee graph 19va sataabdamloo shfley double six aakruteekarana vanti upagraflanu upayoginchi vislaeshimchabadimdi. paripuurakaramaina graph (rendu panktulu vidipoyinappudalla anchutho) nuu shfley graph antaaru.
cubic uparitalaala gurinchi aneka samasyalanu suutram _ 16 muula vyavastha yokka kalayikanu upayoginchi parishkarinchavacchu. udaaharanaku, 27 panktulanu lye group formula _ 16 yokka praathamika praatinidhyam yokka baruvulatho gurtinchavacchu. cubic uparitalampai sambhavinche ekatvaala yokka saadhyamayye samitulanu formula _ 16 ruut system yokka upa vyavasthala paramgaa vivarinchavacchu. eee anusandhaanaaniki oka vivarana emitante, suutram _ 16 jalakamu picard group suutram _ 23 looni anticononical class suutram _ 22 kii orthogonal puurakamgaa udbhavistundi, daani khandana roopamtho (uparitalampai vakrarekhala khandana siddhaantam nundi vastundi). mruduvaina sanklishta cubic uparitalam kosam, picard jalakanni cohomology group formula _ 24 thoo kooda gurtinchavacchu.
eccordt point anedi 27 linelalo 3 kalise point. chaalaa cubic uparitalaalaku eccordt point undadu, conny ituvanti paayintlu anni mruduvaina cubic uparitalaala kutumbaaniki chendina codimension-1 upasamitilo sambhavistaayi.
x" paie cubic uparitalam mariyu saadhaarana sthithilo 6 paayintla vadda suutram _ 4 yokka paeludu madhya gurtimpunu batti, "x" paie 27 panktulanu ilaa chudavachuhu paeladam dwara srushtinchabadina 6 asaadhaaranamaina vakratalu, suutram _ 4 looni 6 paayintla jatala dwara 15 panktula dwaipaakshika parivartanalu mariyu 6 paayintlalo okati minaha annintinee kaligi unna 6 sankuvula dwaipaakshika parivartanalu. icchina cubic uparitalaanni suutram _ 4 yokka paeludugaa okati kante ekkuva margallo chudavachu (vaastavaaniki, 72 ververu margallo), kabatti paeludugaa vivarana mottam 27 rekhala madhya samaruupatanu velladinchadu.
cubic uparitalaalu mariyu suutram _ 16 muula vyavastha madhya sambandham anni dell pejo uparitalaalu mariyu muula vyavasthala madhya sambandhaaniki saadhaaraneekarinchabadutundi. ganita saastramlo aneka aday vargeekaranalalo idhi okati. eee saaruupyatalanu anusaristuu, vera serganova mariyu alexey skorobogatov cubic uparitalaalu mariyu lye group formula _ 16 madhya pratyaksha rekhaganita sambandhaanni icchaaru.
bhouthika saastramlo, 27 panktulanu aaru-dimensional torus (6 momenta; 15 membranelu; 6 fivebranelu) paie m-siddhaantam yokka 27 chaarjeelatoo gurtinchavacchu mariyu samuuham e6 appudu sahajamgaa u-duality groupugaa panichestundi. dell pejo uparitalaalu mariyu toripai emm-siddhaantam madhya unna eee pataanni marmamaina dvandvatvam antaaru.
pratyeka cubic uparitalaalu.
formula _ 1 looni mruduvaina sanklishta cubic uparitalam atipedda automorphism samuuhamtoe fermat cubic uparitalam, dheeni nirvachanam
daani automorphism samuuham 648 kramamlo unna podigimpu suutram _ 32.
tadupari atyanta cemetric smooth cubic uparitalam clebs uparitalam,
suutram _ 33 loe rendu sameekaranaala dwara nirvachinchavacchu.
daani automorphism samuuham 120 kramamlo unna cemetric group formula _ 35. coordinatela sanklishta sarala maarpu tarvaata, clebs uparitalaanni kooda sameekaranam dwara nirvachinchavacchu.
suutram _ 1 loe.
yeka sanklishta cubic uparitalaalo, kylie yokka nodal cubic uparitalam garishta sankhya gala nodlatho kuudina pratyaekamaina uparitalam, 4:
daani automorphism samuuham 24va kramamlo formula _ 39.
nijamaina cubic uparitalaalu.
sanklishtamaina kesuku viruddhamgaa, vaastava sankhyalapai mruduvaina ghana uparitalaala sthalam shaastreeya sthalavignaanamlo anusandhaaninchabadaledu (r yokka sthalavignaanam aadhaaramgaa). daani anusandhaaninchabadina bhagalu (maroe maatalo cheppalante, isotopy varaku mruduvaina nijamaina cubic uparitalaala vargeekarana) ludwig schlafly (1863), felix clain (1865) mariyu hetch dwara nirnayinchabaddaayi. zii. juthen (1875). avi, suutram _ 1loe mruduvaina vaastava ghana uparitalaalu "x" yokka 5 isotopy taragatulu unnaayi, ivi vaastava binduvula suutram _ 41 yokka sthalaakruthi dwara vaeru cheyabadataayi. vaastava binduvula sthalam suutram _ 42kii ledhaa suutram _ 43 mariyu 2-golaala yokka vidadeeyabadina kalayikaku bhinnamainadi, ikkada suutram _ 44 nijamaina prakshepaka samatala suutram _ 45 yokka "r" kaapeela anusandhaaninchabadina mottaanni suuchistumdi. tadanugunamgaa, "x" loe unna vaastava rekhala sankhya 27,15,7,3 ledhaa 3.
oka mruduvaina nijamaina ghana uparitalam daani vaastava binduvula sthalaanni anusandhaaninchinatlayite matrame r paie hetubaddhamainadi, anduvalla munupati aidu kesulalo modati naalugu kesulalo.
x" kosam nirvachinche bahupadi bombierry antargata utpatti dwara prerepinchabadina gassian samishti nundi yaadrucchikamgaa namuunaa cheyabadinappudu "x" paie vaastava rekhala sagatu sankhya suutram _ 46.
cubic uparitalaala madyuli space.
rendu mruduvaina cubic uparitalaalu beejaganita rakaluga isomorphicga untaayi, avi suutram _ 1 yokka konni sarala automorphism dwara samanamga unte matrame. rekhaganita asthira siddhaantam cubic uparitalaala madyuli khaaliini istundi, prati isomorphism taragati mruduvaina cubic uparitalaalaku oka binduvu istundi. eee madyuli space parimaanam 4 kaligi undhi. marinta khachitamgaa, idhi salmon mariyu clebs (1860) dwara baruvunna projective space p (12345) yokka open upasamiti. mukhyamgaa, idhi hetubaddhamaina 4 retlu.
vakratala cone.
beejaganitaparamgaa moosivesina kshetrampai cubic uparitalampai "x" paie unna rekhalanu suutram _ 1 loe "x" ponduparachadaanni suuchimchakumdaa antargatamgaa vivarinchavacchuhu avi sarigga "x" paie unna (−1)-vakratalu, antey suutram _ 12kii isomorphic vakratalu, ivi sweeya-vibhajana-1 kaligi untaayi. alaage, "x" yokka picard latis looni rekhala taragatulu (ledhaa samanamga bhajaka taragati samuuham) khachitamgaa pick ("x") yokka muulakaalu "u", antey suutram _ 50 mariyu suutram _ 51. (idhi suutram _ 1 nundi "x" varaku unna hyperplane line bundle o (1) yokka parimitini anubandha suutram dwara anticononical line bundle suutram _ 22 gaa upayogistundi.)
edainaa projective rakam "x" kosam, vakrarekhala cone antey "x" looni anni vakrarekhalatho vistarinchi unna kumbhakara cone (1-chakraala maadyulo sankhya samaanatvamlooni real vector space formula _ 54loe, ledhaa bays feeled sanklishta sankhyalu ayithe homology group formula _ 55loe). cubic uparitalam kosam, vakrarekhala cone 27 rekhalatho vistarinchi untundi. mukhyamgaa, idhi suutram _ 56loe pedda samaruupata samuuhamtoe kuudina hetubaddhamaina polyhedral cone, suutram _ 16 yokka wheel samuuham. edainaa dell pejo uparitalaaniki vakrarekhala cone gurinchi idhe vidhamaina vivarana undhi.
oka kshetram meeda ghana uparitalaalu.
beejaganitaparamgaa moosiveyabadani kshetram "k" paie mruduvaina ghana uparitalam "x" "k" paie hetubaddhamgaa undavalasina avasaram ledhu. teevramaina sandarbhamgaa, hetubaddhamaina paayintlu lekunda hetubaddhamaina sankhyala q (ledhaa p-adic sankhyala suutram _ 58) paie mruduvaina ghana uparitalaalu unnaayi, eee sandarbhamlo "x" khachitamgaa hetubaddhamainadi kaadu. x" ("k") khaliga lekapothe, beniamino segre mariyu janos kollar dwara "k" paie "x" kaneesam ekakaalikam. k" anantam kosam, ekaarhata anedi "k"-hetubaddhamaina binduvula samiti "x" loe jariski dattamgaa untundani suuchistumdi.
k" yokka sampoorna galois samuuham "k" yokka beejaganita moosiveta suutram _ 59 (suutram _ 16 yokka wheel samuuham yokka konni upa samuuham dwara) paie "x" yokka 27 panktulanu kramabaddheekaristundi. eee charya yokka kontha kakshyalo vidadeeyabadina rekhalu unte, appudu x anedi oka moosivesina binduvu vadda "k" paie "saralamaina" dell pejo uparitalam yokka paeludu. lekapothe, "x" kii picard sankhya 1 untundi. ("x" yokka picard samuuham rekhaganita picard samuuham suutram _ 61 yokka upa samuuham.) taruvaati sandarbhamlo, segre "x" eppuduu hetubaddhamainadi kaadani choopinchindi. marinta balamgaa, yoori manin oka dwaipaakshika drudhatvam prakatnanu niroopinchaaduh picard number 1 thoo rendu mruduvaina cubic uparitalaalu khachitamaina kshetram "k" paie dwaipaakshikamgaa unte mariyu avi isomorphic ayithe matrame dwaipaakshikamgaa untaayi. udaaharanaku, eee phalitaalu q paie aneka cubic uparitalaalanu istayi, avi ekapakshamainavi conny hetubaddhamainavi kaavu.
yeka ghana uparitalaalu.
27 panktulanu kaligi unna mruduvaina cubic uparitalaalaku viruddhamgaa, yeka ghana uparitalaalu takkuva panktulanu kaligi untaayi. antekakunda, vaati saadhaarana roopamlo utpannamayye ekatvam yokka rakanni batti vaatini vargeekarinchavachhu. eee ekatvaalu dainkin rekhaachitraalanu upayoginchi vargeekarinchabaddaayi.
vargeekarana.
suutram _ 65 dwara ivvabadithe, sthaanika akshaamsha suutram _ 64 thoo suutram _ 63 looni saadhaarana yeka ghana uparitala suutram _ 62 saadhaarana roopamlo untundani chebutaaru. suutram _ 65 dwara ivvabadina ekatva suutram _ 62 rakanni batti, idhi suutram _ 65 dwara ivvabadina suutram _ 67 loe projective uparitalaaniki isomorphicga untundi, ikkada suutram _ 69 krindi pattikalo unnatlugaa untundi. antey manam anni yeka ghana uparitalaala vargeekarananu pondavacchu. eee krindi pattika yokka paaraamitulu eee krindi vidhamgaa unnaih suutram _ 70 suutram _ 71 yokka moodu vibhina amsaalu, paaraamitula suutram _ 72 suutram _ 73 loe unnaayi mariyu suutram _ 74 suutram _ 75 yokka oka moolakam. ekatvam suutram _ 76 thoo rendu ververu yeka ghana uparitalaalu unnayani gamaninchandi. 
saadhaarana roopamlo, oka ghana uparitala suutram _ 62 kaneesam oka suutram _ 78 ekatvaanni kaligi unnappudu, adhi suutram _ 80 vadda suutram _ 78 ekatvaanni kaligi untundi. 
yeka ghana uparitalaalapai rekhalu.
yeka ghana uparitalaala vargeekarana prakaram, eee krindi pattika prati uparitalamlo unna rekhala sankhyanu chooputundi.
paaraamitulu laeni yeka ghana uparitalaala automorphism samuuhaalu.
saadhaarana yeka ghana uparitala suutram _ 62 yokka automorphism anedi projective space formula _ 67 nundi formula _ 62 yokka automorphism yokka parimiti. ituvanti automorphisms yeka binduvulanu samrakshistaayi. antekakunda, avi vividha rakaala ekatvaalanu kramabaddhikarinchavu. uparitalam oke rakamaina rendu ekatvaalanu kaligi unte, automorphism vaatini kramabaddheekarinchavachhu. cubic uparitalampai automorphismla sekarana automorphism group anie pilavabade oka samuuhaanni yerparustundi. eee krindi pattika etuvanti paaraamitulu laeni yeka ghana uparitalaala yokka anni automorphism samuuhaalanu chooputundi.

