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Linear series of irregular varieties. (English) Zbl 1094.14502

Ohbuchi, Akira (ed.) et al., Proceedings of the symposium on algebraic geometry in East Asia, Kyoto, Japan, August 3–10, 2001. River Edge, NJ: World Scientific (ISBN 981-238-265-8/hbk). 143-153 (2002).
From the introduction: The purpose of this note is to introduce some application of Fourier-Mukai transform in the study of linear series. To setup, let \(A\) be an abelian variety and \({\mathcal F}\) be a coherent sheaf on \(A\). \({\mathcal F}\) is said to be \(IT^0\) if \(H^i(A,{\mathcal F}\otimes P)=0\) for all \(P\in\text{Pic}^0(A)\) and \(i>0\). By using Fourier-Mukai transform, it is easy to see the following results.
Lemma 1.2. If \({\mathcal F}\neq 0\) is \(IT^0\), then \(h^0(A,{\mathcal F})\neq 0\).
Lemma 1.3. If \({\mathcal F}\) is \(IT^0\) and there is a surjective map \({\mathcal F}\to k(y)\), then the induced map \(H^0(A,{\mathcal F}\otimes P)\to H^0(A,k(y)\otimes P)\) is surjective for general \(P\).
Consider now a variety with \(q(X)>0\). Let \(\text{alb}:X\to\text{Alb} (X)\) be the Albanese map. And let \(a(X)\) be the dimension of image of alb. We say \(X\) is of maximal Albanese dimension if \(a(X)=\dim X\). In general, our idea here can be realized as follows: In order to study the linear series \(|D|\), we consider the push-forward \({\mathcal F}:=\text{alb}_*{\mathcal O}(D)\). If \({\mathcal F}\) satisfies \(IT^0\), then one can obtain some information on base points by Lemma 1.3. For example, we are able to show the following result.
Theorem 1.4. If \(\dim X-a(X)\leq 2\), then \(|7K_X|\) defines a birational map.
In particular, for a 3-fold with \(q>0\), the \(|7K_X|\) defines a birational map.
For the entire collection see [Zbl 1019.00008].

MSC:

14C20 Divisors, linear systems, invertible sheaves
14E05 Rational and birational maps
14J30 \(3\)-folds
14J40 \(n\)-folds (\(n>4\))
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