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

**Read Online or Download Algebraic Curves and Projective Geometry: Proceedings of the Conference held in Trento, Italy, March 21–25, 1988 PDF**

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

**Sample text**

C u r v e s not trivial. n. c u r v e s we have a complete answer to question C Qg, in terms of the numbers s I ..... s l,r I ..... r arising in the is resolution C19, at least when the base field has characteristic O. 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.

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.

Ramita, this subject. with Notati on ~3 = projective 3-space characterististic O; = plane of ~s defined over by an t=O algebraically in some fixed closed field coordinate K of system; 35 Z = s e t of @Z = ideal We d distinct points s h e a f of Z i n ~. Ssp_ If M a minimal we n; resolution reorder I, is in the for ~Z numbers in ~ given s. 's s o j the injection degree matrix [CGO]) the the matrix is of homogeneous in sequence (1); @A = (si-rj). By maximal minors homogeneous ideal If M i s a n y scheme defined If C is a n y and ~C sur j ec t we set (19 have: forms in ~m, is the ideal all and this C2D, of A all its which OM = minors defines then A is persymmetric and its the Hilbert-Burch theorem (see define eH°C~zCnD) a of set of minimal generators Z.

### 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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