
avasheshikkunna intersection

beejaganita jyamithiyil, sheshikkunna intersectionte prashnam inipparayunnava chodikkunnu
intersection oru klaas formula _ 2, intersection uthppannam, oru ambiyant spacinte chau groupil nirnnayikkunnu, eee saahacharyathil, prashnam klaas, sheshikkunna klaas "z" ennu manassilakkuka ennathaanu
ivide formula _ 4 ennaal "z" ll pinthunaykkunna bhagam ennaanu ardhamaakkunnathu; classicalaayi "z" ll pinthunaykkunna bhaagatthinte digriye "z" nte thulyatha ennu vilikkunnu.
enumerative jyaamithiyile prasnangalkkulla parihaarangalaan rund pradhana prayogangal (udha.jii.steynarude conic prashnam), multiple-pooyint formulayude vyulppannam, oru naarile pooyintukal ananthamaayi aduthirikkumbol polum ennaano kanakkaakkaano anuvadikkunna soothravaakyam.
avasheshikkunna intersectionte prashnam pathonpathaam noottaandilethaanu. prashnangaludeyum parihaarangaludeyum aadhunika roopeekaranam fulton macferson moolamaanu. krithyamaayi paranjaal, sheshikkunna intersectionukalude prasnangal pariharikkunnathinulla oru maarggathiloode avaa intersection sidhaantham vikasippikkunnu (athaayath, oru saadhaarana koninte segre klaas oru intersectionil upayogikkunnathiloode.) pathiv ulcherkkal sambandhicha anumaanam durbalamaakunna oru saahacharyathilekkulla oru saamaanyavalkkaranam ithinu kaaranamaakunnu.
nirvachanam.
thaazhepparayunna nirvachanam ithinu kaaranamaan.
anuvadikkuka.
adanja ulcherkkalukalaayirikku, avide "a" oru beejaganita vakabhedavum "z", "w" adanja subskeemukalumaanu. appol, nirvachanam anusarichu, "z" ennathinulla sheshikkunna skeem ithaanu
ivide formula _ 7 projectivisation (classical ardhathil) formula _ 8 aanu anuyojyamaaya sheef nirvachikkunna formula _ 9.
shradhikkuka formula _ 10 ennathu formula _ 12-leeii formula _ 11-nte blow-upp aanenkil, formula _ 13-nuu, sarjakshan formula _ 14 clossed embedding nalkunnu
inclusion formula _ 9 oru saadhaarana ulcherkkalaanenkil ithu isomorphisamaanu.
soothravaakyam _ 17 soothravaakyam _ 18-nte skeem-theoritic connected gadakangalaanenkil,
udaaharanathinu, "y" projective space aanenkil, besouttinte sidhaantham formulayude digri _ 18 formula _ 21 aanennu parayunnu, athinaal intersectionte digriyilekkulla sambhavanakal kanakkakkunnathinulla mattoru maargamaan mukalil paranjirikkunnathu. vaasthavathil, prayogangalil, besouttinte sidhaantham samyojippikkunnu.
soothravaakyam _ 22 adisthaana feeldil verthirichathum parimithamaaya tharatthilumulla skeemukalude pathiv ulcherkkalukalaayirikkedu; udaaharanathinu, "x" i "phalapradamaaya cartier divaisarukalaanenkil (udha.jii., hypersurfaceukal). formula _ 23-nte intersection prodakt
y" enna chau groupile oru gadakamaanu ithu, ithu ingane ezhuthaam
ivide soothravaakyam _ 26 positteev poornnasamkhyakalaanu.
oru settu "kall" nalkiyaal, nammal anuvadikkunnu
soothravaakyangal.
kvillante adhika-intersection formula.
topological krameekaranatthile soothravaakyam kaaranam.
ippol, namukk "y" "" y "" nalkiyittundennu karuthuka, koodaathe "i": "x" = "x" "×" "y" "y" "" y "" ennathu "d" enna codimension regular aanennu karuthuka, angane oraalkk "i" nirvachikkaan kazhiyum! pandattheppole. f" ennathu "i", "i" ennivayude adhika bundle ayirikkatte; athaayath, ithu "i" yude saadhaarana bundle "n" nte bhaagatthinte "x" lekkulla pulbaakkaan. a" "k" − "d" ("x") muthal "a" "k" − "d" ("x") vareyulla homomorphisamaayi naam kaanunna "f"-nte euler klaas (topp chern klaas) ayirikkatte "e" ("f"). appol
evide "njaan"! y" "→" "y" "→" "y" "enna roopeekaranamaanu ithu nirnnayikkunnath".
avasaanamaayi, intersection morphisangal poortthiyaakkunnathinu melpparanja nirmmanavum soothravaakyavum saamaanyavalkkarikkaan kazhiyum; eee vipuleekaranam § 6.6 lum ch lum charcha cheythittundu. 17 ennam. sii. tii.
"proof": oru gissin homomorphisathinte vyakthamaaya roopathil ninnu intersection formula oraalkk kandethaam. r" raankinte "x"-ll "e" oru vekttar bundle ayirikkatte, projective bundle ayirikkatte (ivide 1 ennaal trivial line bundle ennaanu ardhamaakkunnathu). pathivupole, p ("e"), "e" ennivayude oru bhinnamaaya samyojanamaayi nammal p ("e" ±1) yeei thirichariyunnu. appol totological krithyamaaya kramam undu
p ("e" ±1) ll. gissin homomorphism ingane nalkiyittundennu njangal avakaashappedunnu
ivide "e" ("φ") = "c" "r" ("φ") ennathu φ-nte euler classes formula _ 30 ennathu "x" aayi parimithappedutthunna oru gadakavumaanu. kuthivayppu pilarunnathinaal namukk ezhuthaam.
ivide p ("e") ll pinthunaykkunna oru saikkilinte oru klaasaanu "z".
vitni koottukettukalude soothravaakyam anusarichu, namukk "c" ("q" * "e") = (1 − "c" 1 ("o" (1))) "c" ("φ") ennivayum angane undu.
appol namukk labikkum
ivide "s" i "(" e "±1) ennathu" i "-th segre klaas aanu. oru segre klaasinte poojya padam identityum athinte negatteevu padangal poojyavum aayathinal, melpparanja padaprayogam "y" nuu thulyamaanu. aduthathaayi, φ muthal p ("e") vareyulla niyanthranathinu evideyum illaathaakaatha oru vibhaagavum "z" ennathu p ("e") ll pinthunaykkunna oru saikkilinte oru classes aayathinal, athu pinthudarunnu. athinaal, "e" enna projection maappinaayi π ezhuthukayum "e" muthal p ("e" ±1) vare ulppeduthunnathinu "j" ennu ezhuthukayum cheythaal namukk labikkum
randaamathethu muthal avasaanatheth vareyulla samathvam munbatheppole pinthuna kaaranam moolamaanu. ithu gissin homomorphisathinte vyakthamaaya roopathinte thelivu poortthiyaakkunnu.
bakkiyullava aupachaarikavum nerayathum aanu. nammal krithyamaaya kramam upayogikkunnu.
ivide "r" ennathu projection maappaan. vitni summ formula projection formula upayogichu "vii" yude specialisation avasaanippikkunnathin "pii" ennu ezhuthumbol namukk undu
soothravaakyam _ 37
formulayude oru prathyeka cases svayam-intersection formulayanu, athu ingane parayunnu "i": "x" "n" enna saadhaarana bundle upayogichu "y" enna pathiv ulcherkkal nalkiyaal,
(ithu labhikkunnathin, "y" = "y" = "x" edukkuka.) udaaharanathinu, ithil ninnum projection formula ninnum, "x", "y" minusamaarnnathaanenkil, oraalkk formula kanakkakkaam
y" ennathinte chau ringle.
formula _ 40 oru clossed subskeem "x", formula _ 41 asaadhaaranamaaya devisor, formula _ 42 "f" yude niyanthranam ennivayile blow-upp ayirikkatte. f" ennathu oru adacha nimajjanamaayi ezhuthaam, thudarnnu minusamarnna roopeekaranam (udaaharanathinu, "y" ardha-projective aanu). appol, formula _ 43 ll ninnu oraalkk labikkum
udaaharanagal.
udaaharana vibhaagathiludaneelam, adisthaana feeld beejaganithaparamaayi adanjirikkunnu, koodaathe poojyatthinte swabhaavamundu. chuvadeyulla ellah udaaharanangalum (aadyathethu ozhike) ithil ninnullathaanu.
udaaharanam ore ghadakam adangiya rund samathala vakrangalude vibhajanam.
formula 44 umm formula 45 umm formula 46 leeii rund playin karvukalayirikkatte. saidhaanthikamaayi sajjeekarichirikkunna avayude intersection formula 47 oru pooyintinte union ulchertha formula 48 umm aanu. besouttinte sidhaantham anusarichu, eee intersectionil formula 49 pooyintukal adangiyirikkanamennu prathekshikkunnu, kaaranam ithu rund koniksinte intersectionaanu, athinaal eee intersection vyaakhyaanikkunnathin oru sheshikkunna intersection aavasyamaanu. appol formula _ 50
formula _ 51 formula _ 52 ennathu digri formula _ 53 hyperserfaceukalaayathinaal, avayude saadhaarana bundle formula _ 54 nte pulbaakkaan, athinaal rund sheshikkunna gadakangalude numerator formula _ 55 aanu, kaaranam formula _ 56 ennathu vanishing locus formula _ 57 nalkunnu, kaaranam athinte saadhaarana bundle formula _ 58 aanu, athinaal formula _ 59 formula _ 56 aayathinal formula _ 61 aanu. athupole, numeratorum formula _ 62 aanu, athinaal sheshikkunna intersection digri formula _ 62 aanu, formula _ 56 ennathu vanishing locus formula _ 57 nalkunna poornna intersection aayathinal pratheekshikkunnathupole. koodaathe, formulayude saadhaarana bundle formula _ 67 aanu, kaaranam ithu vanishing locus formula _ 68 nalkiyathinaal, soformula _ 69 inverting formula _ 70 seerees 71 formula _ hengving formula naa
udaaharanam moonnu prathalangalile oru vakrathinte alavu.
formula _ 77 moonnu uparithalangalayirikkatte. skeem-theoritic intersection formula _ 78 oru minusamarnna vakram "sii", poojya-dimensional skeem "ennivayude bhinnamaaya samyojanamaanennu karuthuka. oraalkk chodikkaam "s"-nte alavu enthaanu? ithinu #formula enna utharam nalkaam.
udaaharanam anchu varikal nalkunnathinu conix taanjant.
playin konikkukal formula 79 kondu paaraameettar cheiyunnu. anchu pothuvaaya varikalude formula 80 nalkiyaal, formula 81 conix taanjantinte hyperserfaceukalaayiricte; eee hyperserfaceukalkku digri rund undennu kaanikkam.
intersection formula _ 83 ll eratta varakalulla veronese uparithala formula _ 84 adangiyirikkunnu; ithu formula _ 83-nte oru skeem-theoritic connected gadakamaanu.
hyperplane klaas = "z" nte chau ringile "o" (1) nte aadya chern klaas aakuka. ippol, formula _ 87 formula _ 88 formula _ 89 lake pinvaangunnu, athinaal formula _ 90 lekkulla saadhaarana bundle "z" aayi parimithappedutthiyirikku
athinaal, athinte motham chern klaas
athupole, oru saadhaarana formula _ 93-leeii saadhaarana bundle formula _ 94 umm euler sequence aanennu upayogichu, formula _ 95-lekkulla saadhaarana bundle motham chern klaas aanennu namukk labikkum.
athinaal, formula _ 95-nte segre klaas ithaanu
athinaal, "z"-nte thulyatha ithaanu
besouttinte sidhaanthamanusarichu, formula _ 83nte digri formula _ 101 aanu, athinaal sheshikkunna ganathil thannirikkunna anchu varikalodum oru advitheeya conic taanjantumaayi poruthappedunna orotta pooyint adangiyirikkunnu.
pakaramaayi, "z"-nte thulyatha #formula upayogichu kanakkakkaam.; formula _ 102 umm formula _ 103 umm aayathinal, athu ithaanu
udaaharanam anchu konikkukal nalkunnathinu conix taanjant.
pothuvaaya sthaanangalil namukk anchu playin conix formula 105 nalkiyittundennu karuthuka. muun udaaharanathilennapole oraalkk krithyamaayi munnott pookaam. athinaal, formula _ 106 ennathu formula _ 107 lekkulla conix taanjantinte hypersurface ayirikkatte; athinu digri 6 undennu kaanikkam. intersection formula _ 108 ll eratta varikalude veronese uparithalam "z" adangiyirikkunnu.
udaaharanam oru shudheekaricha gissin homomorphisathinte nirmmaanatthinte pravarthanakshamatha.
rund pathiv ulcherkkal soothravaakyam nalkumbol, 109 enna vibhaagathinte thalakkettaanu faktoriyaalitti.
samathwathinu inipparayunna ardhamunda