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Boundedness of log-pluricanonical maps for surfaces of log-general type in positive characteristic. (English) Zbl 1483.14066

Summary: In this article we prove the following boundedness result: Fix a DCC set \(I\subseteq [0, 1]\). Let \(\mathfrak{D}\) be the set of all log pairs \((X, \Delta)\) satisfying the following properties: (i) \(X\) is a projective surface defined over an algebraically closed field, (ii) \((X, \Delta)\) is log canonical and the coefficients of \(\Delta\) are in \(I\), and (iii) \(K_X+\Delta\) is big. Then there is a positive integer \(N=N(I)\) depending only on the set \(I\) such that the linear system \(|\lfloor m(K_X+\Delta)\rfloor|\) defines a birational map onto its image for all \(m\geq N\) and \((X, \Delta)\in\mathfrak{D} \).

MSC:

14J29 Surfaces of general type
14E05 Rational and birational maps
14E30 Minimal model program (Mori theory, extremal rays)
14G17 Positive characteristic ground fields in algebraic geometry
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References:

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