By E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco Ghione, Ferruccio Orecchia (eds.)
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Additional info for Algebraic Geometry — Open Problems: Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982
Of the k - v e c t o r of V the i n t e g e r of V is Pm(V) variety V we space Hi(v,F), q(V) = h I ( O v) . = h O ( ~ v~m), where ~V sheaf. We d e n o t e by ~(V) the K o d a i r a d of V and by X(O v) = ~ (-l)ihi(c9 v) the Euleri=O characteristic of V. let sr(~PV) be the r - s y m m e t r i c p-forms closed we m e a n a c o m p l e t e F on a d - d i m e n s i o n a l the m - p l u r i ~ e n u s dimension Poincare By v a r i e t y sheaf results fix an a l g e b r a i c a l l y zero. the d i m e n s i o n is the c a n o n i c a l and p r e l i m i n a r y Finally, tensor for all p, O < p < d, of the sheaf of r e g u l a r ~P.
F , i , T x = b . ~ , T x = 2b. f , ( f * 2 a - i , T x) that torsion all s c C of X along a n d b y Y' p':Y' ÷ C be the r e s t r i c t i o n fibre p'-l(s), C of the (C) o f s i n g u l a r e : C ÷ Y fibre b y e:X' ÷ X by E the exceptional transform of -i of t h e We denote the proper s £ C, Y = f section point e(C), curve foe is i s o m o r p h i c of Y. to Y' Let The to the b l o w i n g - u p 48 of X in its s singular point. p' Furthermore factorizes p,~ through p ! Y' -" C C where p double etale are is a f i b r a t i o n covering nonsingular intersection projection with in p r o j e c t i v e of C Moreover transversal can be i d e n t i f i e d p.
De F a n o " s ' a p p r o c h e en e s t assertions ~n~ri~ue irrationnelle l'espace de d i m e n - : rationnelle, ou G e s t rement I1 doubles de c o d i m e n s i o n tordue" On r e n c o n t r e de de rationnelle des vari~t~s ~videmment unirationnelle. au § 9 u n e v a r i ~ t ~ le int~ressant irrationnelle vari~t~ cro~t. £ ces en dimension est s~rie en dimension probl~me X× ~1 Elle la ; conjecturalement Signalonsun Si d'une degr~ precis cubi~ue famille rationnelles plus dans ? de p o i n t s En p a r t i c u l i e r ~u'une serait cas, : d~une vari~ t~ que quand Donner des Prouver I1 de F a n o .
Algebraic Geometry — Open Problems: Proceedings of the Conference Held in Ravello, May 31 – June 5, 1982 by E. Ballico, Ph. Ellia (auth.), Ciro Ciliberto, Franco Ghione, Ferruccio Orecchia (eds.)