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The 5-canonical system on 3-folds of general type. (English) Zbl 1121.14029
The study of pluricanonical maps is central in the biregular theory of algebraic varieties of general type. It is particularly interesting to know which pluricanonical map is birational onto the image. The paper under review studies minimal Gorenstein 3-folds with canonical singularities. The main result is that under this hypothesis the 5-th canonical map is birational.
The result is sharp and improves on all previously known bounds in the literature. The result is obtained by a subtle analysis of pluricanonical systems of 3-folds and surfaces. It is interesting to note that, like in the surface case, the bicanonical map is somewhat directing the proof. It has been recently proved that for varieties of general type there is a function depending only on the dimension that bounds the birationality of pluricanonical maps [C. D. Hacon and J. McKernan, Invent. Math. 166, No. 1, 1–25 (2006; Zbl 1121.14009), S. Takayama, Invent. Math. 165, No. 3, 551–587 (2006; Zbl 1108.14031)].

MSC:
14J10 Families, moduli, classification: algebraic theory
14E25 Embeddings in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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