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Polarized surfaces \((X,L)\) with \(g(L)=q(X)+m\) and \(h^0(L)\geq m+2\). (English) Zbl 1051.14044

The paper deals with a problem of classification of surfaces which is in some sense related to Fujita’s conjecture: Let \((X,L)\) be a polarized manifold, then \(g(L)\geq q(X)\) (here \(g(L)\) and \(q(X)\) are sectional genus and irregularity of \((X,L)\)).
From previous results in case \(\dim X = 3\) [Commun. Algebra 28, 5769–5782 (2000; Zbl 1023.14002)], the author is lead to consider the following problem: classify polarized manifolds \((X,L)\) with \(\dim X = n\), \(g(L)= q(X)+m\) and \(h^0(L)\geq m+n\). The case \(n=2\) is treated here: the main result is that when a polarized surface \((X,L)\), with \(g(L)= q(X)+m\) has \(h^0(L)\geq m+2\), then its Kodaira dimension is \(\kappa (X) = -\infty\). Other results are given in particular cases (e.g. if \(X\) is minimal, \(\kappa (X) = -\infty\) and \(q(X)=0\), then it is shown that \(h^0(L)>m+2\)).

MSC:

14J25 Special surfaces
14C20 Divisors, linear systems, invertible sheaves

Citations:

Zbl 1023.14002
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References:

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