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Additional resources for Algebraic Curves and Projective Geometry: Proceedings of the Conference held in Trento, Italy, March 21–25, 1988
Theorem Let Z be a set of d distinct points in the plane ~ s over a field K with charCK)=O. Ss _1 arithmetically for n = 1 ..... and rIS'''~ru" normal Then curve C~ s n > r n _ 2. u-e s such there that exists Z = Cc~ a if smooth and only if Note that since the curve C above is smooth and arithmetically normal, it is also irreducible. Observe that when Z is complete intersection i n ~, t h e n ~ = 2 s o the condition of the theorem is empty and we get back the well known fact that a complete intersection set of points in ~ is the plane section of a smooth arithmetically normal Cin fact, complete intersection9 curve.
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.
LR] R. Lazarsfeld and P. Rao, Linkage of general curves of large degree . In: Algebraic Geometry- open problems (Ravello 1982). Lect. Notes Math. 997, 267-289. Berlin, Heidelberg, New York: Springer 1983. [HH] R. Hartshorne and A. Hirschowitz, Smoothing algebraic space curves. In: Algebraic Geometry, Sitges (Barcelona 1983). Lect. Notes Math. 1124, 98-131. Berlin, Heidelberg, New York: Springer 1983. [M1] J. Migliore, On Linking double Lines, Trans. AMS 294 (1986), 177-185. [M2] J. Migliore, Liaison of a Union of Skew Lines in p4, Pac.