On structure of small contractions of odd dimensional projective varieties

Abstract

Let X be a smooth projective variety of dimension 2k − 1 (k ⩾ 3) over the complex number field. Assume that fR: X → Y is a small contraction such that every irreducible component Ei of the exceptional locus of fR is a smooth subvariety of dimension k. It is shown that each Ei is isomorphic to the k-dimensional projective space Pk, the k-dimensional hyperquadric surface Qk in Pk+1, or a linear Pk−1-bundle over a smooth curve.