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By E. Arrondo, M. Pedreira, I. Sols (auth.), Edoardo Ballico, Ciro Ciliberto (eds.)

ISBN-10: 3540481885

ISBN-13: 9783540481881

ISBN-10: 3540515097

ISBN-13: 9783540515098

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Extra resources for Algebraic Curves and Projective Geometry: Proceedings of the Conference held in Trento, Italy, March 21–25, 1988

Example text

130 (1987), 153-170. [S] P. Schenzel, Notes on liaison and duality, J. Math. , 22- 3 (1982) 485-498. [Sw] P. D. thesis, Brandeis University (1982). PLANE SECTIONS NORMAL OF ARTHMETI (ALLY CURVES I N ~s Luca Chi anti ni Ferruccio Or e c c h i a Dipartimento di M a t e m a t i c a U n i v e r s i t ~ di Napoli. g. Z is i n " u n i f o r m p o s i t i o n " ( s e e [HID, i t s n u m e r i c a l c h a r a c t e r is c o n n e c t e d ( s e e [GP]) a n d s o on. O n t h e o t h e r h a n d , if w e h a v e more precise informations o n t h e g e o m e t r i c p r o p e r t i e s of t h e c u r v e C, w e w o u l d l i k e t o d e s c r i b e m o r e c a r e f u l l y w h i c h " k i n d " of s e t of points Z may arise cutting C with any (even not general9 plane; a d e e p e r k n o w l e d g e of t h o s e s e t s of p o i n t s s h o u l d b e u s e f u l i n t h e classification of s p a c e curves.

We k n o w t h a t t h i s is t r u e for u=3. For g e n e r a l u, c o n s i d e r t h e m a t r i x gCND o b t a i n e d b y fCM9 e r a s i n g t h e f i r s t a n d l a s t column. It is c l e a r that CaCgCMDDm,m+l>0 for any m so we may apply the - inductive hypothesis on gCMg, further by the description of the 39 components i n WCP) w e k n o w t h a t f o r fCMD g e n e r a l i n W P ~ Y C g C N O ) ; t a k i n g gCMD s u c h t h a t P ~ Y C g C N O D , it f o l l o w s t h a t fop fCNO g e n e r a l in W, i t s m i n i m a l m i n o r is s m o o t h at P, f u r t h e r t h e m a x i m a l m i n o r s s e p a r a t e t a n g e n t p l a n e s at P; t a k i n g gCNO s u c h t h a t P ~ Y C g C N O ) , we get YCfCNO) complete intersection, at P, of a s m o o t h s u r f a c e a n d a s u r f a c e n o t t a n g e n t t o t h e f i r s t one; s o w e a r e done.

Given a. n. n. n. c u r v e cutting Z on ~ is a question on the smoothability of C(Zg: we want a smoothing of CC2D by a flat family which fixes the p o i n t s o f Z. n. c u r v e s C w a s s t u d i e d by Peskine 34 and Szpiro Cwho proved i n [PSI t h a t C i s s m o o t h a b l e when all numbers s i-rj are positiveD and by Sauer C[S]D who solved problem sequence showing that C is smoothable if and only i f, to the C29 above, s n > _ r n _ 2 f o r all n. n. c u r v e , h o w e v e r any set Z to a set of points in generic over 7 can such smoothing position.

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Algebraic Curves and Projective Geometry: Proceedings of the Conference held in Trento, Italy, March 21–25, 1988 by E. Arrondo, M. Pedreira, I. Sols (auth.), Edoardo Ballico, Ciro Ciliberto (eds.)


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