\(E1\)-blowing ups of terminal singularities
Let \(o\in Z\) be a 3-dimensional terminal singularity. A plt-blowup is a projective birational morphism \(f:X \to Z\) with an irreducible exceptional divisor \(S\) such that \((X,S)\) is purely log terminal. The exceptional divisor \(S\) is a normal surface, and there is an adjunction formula \((K_X+S)| _S = K_X+\text{Diff}_S\), where \(\text{Diff}_S\) is a naturally defined \(\mathbb{Q}\)-Weil divisor on \(S\). The pair \((S,\text{Diff}_S)\) is klt and \(-(K_S+\text{Diff}_S)\) is ample over \(f(S)\). This leads to the question of classifying the following situations: \(f(S)\) is a point, \(S\) is projective and \((S,\text{Diff}_S)\) is a log del Pezzo surface. \(f(S)\) is a curve and \(S/f(S)\) is a log conic bundle. In previous work, \textit{Yu. G. Prokhorov} and \textit{V. V. Shokurov} [Izv. Math. 65, 1169--1196 (2001); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 65, No. 6, 99--128 (2001; Zbl 1068.14018)] have raised the question on the classification of singularities and Mori contractions in terms of complements. Shokurov's proposal to subdivide all regular 2-dimensional complements into the seven types \(A1\), \(E1\), \(D2\), \(E2\), \(E3\), \(E4\) and \(E6\) is parallel to the Kodaira-classification of singular fibres on an elliptic fibration. Problem: Find the relation between the classification of terminal singularities and the classification of 2-dimensional regular complements. In the present paper, the author studies the case \(E1\), which is an analog of the Kodaira-type-\(I_0\) fibres. Main results: Let \(o\in Z\) be as above and \(f:(X,S)\to Z\) be a plt-blowup. Assume that \((S,\text{Diff}_S)\) has a complement of type \(E1\) and no complements of types \(A1\), \(D2\). (i) \(S\) is a projective del Pezzo surface with DuVal singularities, i.e. \(f(S)=o\). (ii) \(\text{Diff}_S= (1-1/m)C\), where \(2\leq m \leq 6\) and \(C\) is a smooth elliptic curve contained in the smooth locus of \(S\). (iii) \(X\) has only terminal singular points outside \(C\) and a singularity locally isomorphic to \(\mathbb C^2/\mu_m(1,q) \times \mathbb C^1\) near each point \(P\in C\), where \(\text{gcd}(m,q)=1\). (iv) There is a complete list given of 13 possible cases. The complements are minimal in the sense of the appropriate ordering \(A1\leq D2\leq E1\leq \dots \) Except for one of the cases, the blowup \(f\) is \((K_Z+F)\)-crepant: \(K_X+F_X=f^*(K_Z+F)\), where \(F\in | -K_Z| \) is a general member and \(F_X\) the proper transform of \(F\). Moreover, the dual graphs of surface contractions \(F_X\to F\) are listed for those cases.
scientific article; zbMATH DE number 1982004