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Quartic-canonical systems on canonical threefolds of index 1. (English) Zbl 1083.14518

Summary: Let \(X_0\) be a projective canonical threefold with an ample Cartier canonical divisor \(K_{X_0}\). It is now known that \(|mK_{X_0}|\) is base point free for \(m\geq 5\). The aim of this paper is to prove the following
Theorem. Let \(X_0\) be a projective canonical threefold with an ample Cartier canonical divisor \(K_{X_0}\). Then \(|4K_{X_0}|\) is base point free.
This is certainly the expected one in view of the strong Fujita conjecture, because \(K^3_{X_0}\) is always even. The difficulty comes in when we have to deal with singular points of \(X_0\) which are not cDV. There is no uniform bound on multiplicity of \(X_0\) at such points. So the usual multiplier ideals method would not work. We avoid this by choosing an effective divisor from \(|2K_{X_0}|\) instead of using Riemann-Roch theorem, to create the first critical variety.

MSC:

14J30 \(3\)-folds
14C20 Divisors, linear systems, invertible sheaves
Full Text: DOI

References:

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