The geometry of some special algebraic varieties

نویسنده

  • Alessandra Sarti
چکیده

Let G ⊂ S O (4 ) d e n o te a fi n ite su b g ro u p c o n ta in in g th e H e ise n b e rg g ro u p . In th e se n o te s w e c la ssify a ll th e se g ro u p s, w e fi n d th e d im e n sio n o f th e sp a c e s o f Gin v a ria n t p o ly n o m ia ls a n d w e g iv e e q u a tio n s fo r th e g e n e ra to rs w h e n e v e r th e sp a c e h a s d im e n sio n tw o . Th e n w e c o m p le te th e stu d y o f th e c o rre sp o n d in g G-in v a ria n t p e n c ils o f su rfa c e s in P3 w h ich w e sta rte d in [S]. It tu rn s o u t th a t w e h a v e fi v e m o re p e n c ils, tw o o f th e m c o n ta in in g su rfa c e s w ith n o d e s. 0. In tro d u ctio n C o n sid e r th e K le in fo u r g ro u p V ⊆ S O(3 ). L e t Ṽ d e n o te its in v e rse im a g e in S U (2 ) u n d e r th e u n iv e rsa l c o v e rin g S U (2 )→S O(3 ). T h e im a g e o f th e d ire c t p ro d u c t Ṽ × Ṽ in S O(4 ) u n d e r th e d o u b le c o v e rin g S U (2 )×S U (2 )→S O(4 ) is th e H e ise n b e rg g ro u p H. In th is n o te w e c la ssify a ll th e su b g ro u p s G o f S O(4 ) w h ich c o n ta in H. F irst w e c la ssify a ll th e su b g ro u p s o f S U (2 ) × S U (2 ) w h ich c o n ta in Ṽ × Ṽ , th e n th e ir im a g e s in S O(4 ) a re th e g ro u p s G (c f. p ro p o sitio n 1 .1 a n d se c tio n 1 .4 ). T h e y o p e ra te in a n a tu ra l w a y o n C[x0, x1, x2, x3], th e rin g o f p o ly n o m ia ls in fo u r v a ria b le s w ith c o m p le x c o e ffi c ie n ts. In se c tio n 3 w e g iv e g e n e ra to rs fo r th e sp a c e s C[x0, x1, x2, x3] G j o f h o m o g e n e o u s G-in v a ria n t p o ly n o m ia ls o f d e g re e j w h e n e v e r th is d im e n sio n is tw o . S in c e th e g ro u p s G c o n ta in H, w e h a v e in v a ria n t p o ly n o m ia ls o n ly in e v e n d e g re e . W h e n th e d im e n sio n is tw o th e g e n e ra to rs a re th e m u ltip le q u a d ric qj/ 2 = (x 0 + x 1 + x 2 + x 3 )j/ 2 (triv ia l in v a ria n t) a n d a n o th e r p o ly n o m ia l o f d e g re e j w h ich w e d e n o te b y f . T h e p e n c ils f + λ q 2 = 0, λ ∈ P1, o f su rfa c e s in th e th re e d im e n sio n a l c o m p le x p ro je c tiv e sp a c e P3 h a v e th e n a la rg e sy m m e try g ro u p (th is is th e re a so n w h y w e c o n sid e r ju st su b g ro u p s o f S O(4 ) c o n ta in ig H). W e d e sc rib e th e m in se c tio n 4 . In p a rtic u la r w e fi n d th e sin g u la r su rfa c e s c o n ta in e d in it. In [S ] w e c o n sid e re d th e c a se o f G = TT , OO, II w h ich a re th e im a g e s in S O(4 ) o f th e d ire c t p ro d u c ts T̃ × T̃ , Õ × Õ, Ĩ × Ĩ w h e re T̃ d e n o te s th e b in a ry te tra h e d ra l g ro u p , Õ th e b in a ry o c ta h e d ra l g ro u p , Ĩ th e b in a ry ic o sa h e d ra l g ro u p in S U (2 ). W e d e n o te d th e g ro u p s th e re b y G6, G8 a n d G12 a n d w e c a lle d th e m b i-p o ly h e d ra l g ro u p s. W e fo u n d p e n c ils c o n ta in in g su rfa c e s w ith m a n y n o d e s (=o rd in a ry d o u b le p o in ts). In p a rtic u la r th e d e g re e tw e lv e II-in v a ria n t p e n c il c o n ta in s a su rfa c e w ith 6 00 n o d e s w h ich im p ro v e s th e p re v io u s lo w e r b o u n d fo r th e m a x im a l n u m b e r o f n o d e s o f a su rfa c e o f d e g re e tw e lv e in P3 (c f. [C ]). H e re w e d e sc rib e th e o th e r G-in v a ria n t p e n c ils a n d sh o w th a t w e h a v e tw o m o re p e n c ils w h ich c o n ta in su rfa c e s w ith n o d e s (th e o th e rs d o n o t c o n ta in su rfa c e s w ith iso la te d sin g u la ritie s a t a ll). W e list th e g ro u p s G a n d th e d e g re e s j b e lo w , a s w e ll a s th e n u m b e r o f n o d e s o n th e sin g u la r su rfa c e s. In e a ch p e n c il w e h a v e fo u r o f th e se sin g u la r su rfa c e s a n d th e n o d e s th e re fo rm ju st o n e G-o rb it. F o r th e c o n v e n ie n c e o f th e re a d e r w e re c a ll th e re su lts a b o u t th e TT -, OO-, a n d II-in v a ria n t p e n c ils to o .

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تاریخ انتشار 2007