Two theorems on elementary contractions
Let \(X\) be a nonsingular projective variety over \(\mathbb{C}\), with \(n=\dim X \geq 4\). Let \(L\) be an ample line bundle over \(X\). Here we study projective maps of \(X\), \(f: X \to Y\), associated to the linear system of \(m(K_ X+rL)\), where \(r\) and \(m\) are positive integers, \(m\) sufficiently large (hence \(f\) has connected fibers and \(Y\) is normal). In particular we prove: Theorem A. Let \(f:X \to Y\) be an elementary small contraction whose good supporting divisor is \(K_ X+(n-3)L\). Then the exceptional locus \(E\) of \(f\) is a disjoint union of its irreducible components \(E_ i\) \((i=1,\dots,n)\) such that \(E_ i \cong \mathbb{P}^{n-2}\) and \(N_{E_ i/X} \cong {\mathcal O}_{\mathbb{P}^{n-2}} (-1)^{\oplus 2}\). Moreover there exists a flip \(f^ +:X^ + \to Y\) of \(f\) from a nonsingular projective variety \(X^ +\). This is a generalization to higher dimension of the case \(n=4\) proved by \textit{Y. Kawamata} [Math. Ann. 284, No. 4, 595-600 (1989; Zbl 0668.14008)]. In the case \(L\) is very ample the theorem was proved in a completely different way by \textit{M. Andreatta}, \textit{E. Ballico} and \textit{J. Wiśniewski} [in Classification of irregular varieties, minimal models and abelian varieties, Proc. Conf. Trento 1990, Lect. Notes Math. 1515, 1-11 (1992; Zbl 0787.14034)]. Theorem B. Let \(f:X \to Y\) be a contraction map onto a \(d\)-dimensional normal variety \(Y\) whose good supporting divisor if \(K_ X+rL\), where \(r=n-d\). Suppose that \(f\) is an elementary contraction and that it is equidimensional. Then \(Y\) is smooth, \({\mathcal E}:=f_ *L\) is a locally free sheaf of rank \(r+2\) and \(X\) embeds over \(Y\) into \(\mathbb{P} ({\mathcal E})\) as a divisor of relative degree 2. For further results similar to theorem \(B\) (case \(m=2\) and without the hypothesis of elementary, equidimensional contraction) we refer to (2,3) in the paper by \textit{M. Beltrametti} and \textit{A. J. Sommese} [J. Math. Soc. Japan 43, No. 2, 381-412 (1991; Zbl 0754.14027)] and to the Ph. D. thesis of \textit{G. M. Besana} (Notre Dame Univ).
scientific article; zbMATH DE number 537343
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