On small contraction of higher dimensional projective varieties (Q2725226)

The author considers small contraction of extremal rays. Let \(f : X \to Y\) be a small contraction of an extremal ray from the nonsingular projective 4-fold \(X\). In Math. Ann. 284, No. 4, 595-600 (1989; Zbl 0661.14009), \textit{Y. Kawamata} proved that every irreducible component of the \(f\)-exceptional set is isomorphic to the projective plane. The author generalizes this result as: Let \(f : X \to Y\) be a small contraction of an extremal ray which equals the set consisting of effective 1-cycles perpendicular to the nef divisor \(K_X + (n-k)A\), where \(A\) is an ample divisor on the non-singular projective variety \(X\) of dimension \(n\), where \(0 < k < n\) is an integer. Let \(F\) be an arbitrary fibre of \(f:E \to f(E)\) where \(E\) is an irreducible component of the \(f\)-exceptional set. Suppose that \(\dim F < n-k+2\). Then every irreducible component of \(F\) is isomorphic to the projective \((n-k+1)\)-space. As a consequence, the author recovers Kawamata's result above by letting \(A = mH - K\) for some \(m > 0\) where \(H\) is nef and perpendicular to the extremal ray defining \(f\). Another consequence is: Let \(f : X \to Y\) be a small contraction of an extremal ray which equals the set consisting of effective 1-cycles perpendicular to the nef divisor \(K_X + (n-3)A\), where \(A\) is an ample divisor on the nonsingular projective variety \(X\). Then every irreducible component of the \(f\)-exceptional set is isomorphic to the projective \((n-2)\)-space. The proofs use results by \textit{M. Andreatta, E. Ballico} and \textit{J. A. Wisniewski} [Math. Ann. 297, No. 2, 191-198 (1993; Zbl 0789.14011)] and by \textit{M. Andreatta} [Math. Ann. 300, No. 4, 669-679 (1994; Zbl 0813.14009)].