Minimal model theory for log surfaces (Q435222)

A log surface is a pair \((X,\Delta)\) such that \(X\) is a normal projective surface over \(\mathbb{C}\) and \(\Delta\) is a \(\mathbb{Q}\)-divisor on \(X\) whose coefficients are between 0 and 1.NEWLINENEWLINEWhile the classical classification theory of smooth projective surfaces, essentially due to the Italian school, somehow represents the base of the recent Mori theory and of the minimal model program in higher dimension, in this paper a modern minimal-model-theoretic approach is used to achieve new results about the birational geometry of log surfaces, that in particular generalize some of the theorems in [\textit{T. Fujita}, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30, 685--696 (1984; Zbl 0543.14004)].NEWLINENEWLINEMore precisely, by using his generalized cone and contraction theorem see [\textit{O. Fujino}, Publ. Res. Inst. Math. Sci. 47, 727--789 (2011; Zbl 1234.14013)], the author proves the existence of minimal models, the abundance conjecture and the finite generation of the log canonical ring for every \(\mathbb{Q}\)-factorial log surface \((X,\Delta)\), without any assumption on the singularities of the pair.