Supplement to classification of threefold divisorial contractions
Let \(f: Y\rightarrow X\) be a \(K_Y\)-negative extremal contraction of a normal \(\mathbb{Q}\)-Gorenstein terminal threefold of relative Picard number \(\rho(Y/X)=1\). The map \(f\) is projective with connected fibers and \(-K_Y\) is \(f\)-ample. Assume that it contracts a prime divisor \(E\subseteq Y\) to a point \(P\in X\), then \(E=\)Exc\((f)\) and \(K_Y=f^*K_X+a(E,X)E\) for some rational number \(a(E,X)>0.\) A map like is indispensable in the minimal model program of terminal threefolds. From the general theory of singularities in the minimal model program, we know that \(P\in X\) is normal, \(\mathbb{Q}\)-Gorenstein, and terminal. Three dimensional terminal singularities were classified by \textit{S. Mori} [Nagoya Math. J. 98, 43--66 (1985; Zbl 0589.14005)].NEWLINENEWLINEIn the paper under review, the author studies the case where \((X,P)\) is of type \(e1\) with \(P=cD/2\) and \(a(E,X)=2\) via the technique developed in [\textit{M. Kawakita}, Duke Math. J. 130, No. 1, 57--126 (2005; Zbl 1091.14008)]. Combined with \textit{M. Kawakita}, [Duke Math. J. 130, No. 1, 57--126 (2005; Zbl 1091.14008)] and \textit{T. Hayakawa}, [Publ. Res. Inst. Math. Sci. 35, No. 3, 515--570 (1999; Zbl 0969.14008); Publ. Res. Inst. Math. Sci. 36, No. 3, 423--456 (2000; Zbl 1017.14006)], the author completes the classification of the \(K_Y\)-extremal divisorial contraction \(f: Y\rightarrow X\) of \(\rho(Y/X)=1\) that contracts the exceptional divisor \(E\) to a non-Gorenstein point \(P\). As a corollary, every such a contraction is a weighted blowup.NEWLINENEWLINEOn the other hand, if \(P\in X\) a Gorenstein point, in [\textit{A. Corti} (ed.) and \textit{M. Reid} (ed.), Explicit birational geometry of 3-folds. London Mathematical Society Lecture Note Series. 281. Cambridge: Cambridge University Press. (2000; Zbl 0942.00009)] it was shown that \(f\) is a weighted blow up if \(P\in X\) is a rational double point. Later on, the author showed that a contraction is a weighted blow up if \(P\in X\) is a smooth point [Invent. Math. 145, No. 1, 105--119 (2001; Zbl 1091.14007)], an \(A_1\) point [Compos. Math. 133, No. 1, 95--116 (2002; Zbl 1059.14020)], or a \(cA_n\) singularity for any \(n\geq1\), [J. Am. Math. Soc. 16, No. 2, 331--362 (2003; Zbl 1091.14008)]. To determine in general if \(f\) is a weighted blowup, [J. Am. Math. Soc. 16, No. 2, 331--362 (2003; Zbl 1091.14008)], the remaining unknown cases are the cases where \((P\in X)\) is a \(cA_2\) point that has \(a(E,X)=3\), or a \(cD_n (n\geq1)\) or \(cE_n (n=6,7,8)\) point with \(a(E,X)\leq4\). However, the author suspects whether the same method of the paper under review would settle all the remaining cases.
scientific article; zbMATH DE number 6050955