By Enrico Arbarello, Corrado De Concini (auth.), Eduard Casas-Alvero, Gerald Welters, Sebastian Xambó-Descamps (eds.)
Read Online or Download Algebraic Geometry Sitges (Barcelona) 1983: Proceedings of a Conference held in Sitges (Barcelona), Spain October 5–12, 1983 PDF
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Additional resources for Algebraic Geometry Sitges (Barcelona) 1983: Proceedings of a Conference held in Sitges (Barcelona), Spain October 5–12, 1983
Pn } is separately: it suffices = O, n We two cases such sections linear b) of forms again r that the W are has k rank linear on Pl'""Pk defined only up 65 to non-zero scalar multiples, We set further V = H°(R,OR(k-I)), tion Pk+i and R ~ p1. The statement that, then the b i = evaluation c i = evaluation at such but this does not matter for our purposes). ,e k e = (e I ..... ek, i ...... i), isomorphism. In particular then = 0. D. Corollary 19. Let C be the union n is at least 3, there exist Proof.
L to the normaliza- C to 3, and L is generic as in D. e. C,D nodes). the datum of a proper flat family such that T is a smooth curve, S is a reduced divisor in ced by the projection of ii) the integral good is not t r i v i a l A degeneration f: to be denote is not t r i v i a l . We r e m a r k def. I. onto such that p 2 × T, f is indu- T f-l(to)-~C, f-l(tl)-----D 59 the fibration iii) exactly f is locally at a finite s e t of n o d e s the isomorphism provided The tion nodes (according Definition to the of C ~.
Pd to S e v e r i A nodal terminology to referred a nodal to a s Definition degree n where ~C A B. C ~ D out sheaf: in this directly more generic even if Definition 6. irreducible with main stream riety of CI union C2 = 0, curve of is a (here C2 are C1 of except ( h e r e we a r e smooth points nodes of of t h e free on condition using S. degenera- exists different a a reduced plane C being a is often curve C [Ba], C be of if [Be] ). F is an inver- thetacharacteristic H°(C,F) to (D F ~- H o m o c ( F , ° ~ C ) , (cf.