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Base curves of multicanonical systems on threefolds

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Base curves of multicanonical systems on threefolds COMPOSITIO MATHEMATICA P. M. H.WILSON Base curves ofmulticanonical systems on threefolds Compositio Mathematica, tome 52, no 1 (1984), p. 99-113. <> © Foundation Compositio Mathematica, 1984, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 99 BASE CURVES OF MULTICANONICAL SYSTEMS ON THREEFOLDS P.M.H. Wilson Compositio Mathematica 52 (1984) 99-113 © 1984 Martinus Nijhoff Publishers, The Hague. Printed in The Netherlands Introduction Let V be a smooth complex projective threefold of general type and K v the canonical divisor on V. In this paper we consider in detail the case when there exists an integer m &#x3E; 0 such that the m-canonical system |mKv| has no fixed components, and the corresponding rational map ~mKV is generically finite; in particular we study the base locus of this linear system. Using the theory of Hilbert schemes, it is easy to see that in this case KV is arithmetically effective (denoted a.e.); i.e. KV · C 0 for all curves C on V. In [12] (Theorem 6.2) we saw that in the case when Kv is a.e., the canonical ring R(V) = ~ n0 H0(V, nKv) is finitely generated as an algebra over the complex numbers if and only if the linear system |nKV| has no fixed points for some n &#x3E; 0. It is easy to see that if K v. C &#x3E; 0 for every base curve C of 1 mKv then such an n does exist (1.2). Thus the interesting base curves C are those with KV’ C = 0. After this paper was written, the

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