Structure Of The Minimal Dimensional Exceptional Locus Of Small Contraction Over Projective Varieties Of Odd Dimension

One of the main research fields in algebraic geometry is the birational equivalent classifications of higher dimensional algebraic varieties, of which the key problem is to construct minimal models through contractions of algebraic varieties. Let X be an n dimensional projective variety. If X is not a minimal model,there exists a surjective morphism f:X→Y from X onto a normal projective variety Y. SetE = {x ∈ X | f is not local isomorphism at x}is called the exceptional locus of f. Structure of f is mainly determined by thestructure of E.f:X→Y is called a small contraction if dimE ≤dim X - 2. The critical step for constructing the minimal model is to make clear the structure of the exceptional locus E of small contraction f:X→Y.In 1988, Mori([14]) obtained a complete classification of E in the case that dim X = 3, and from which Mori proved that there exists a minimal model in a birational equivalent class of non-uniruled projective varieties.After that, Kawamata and Zhang stated that structure of E for the case that dim X = 2k and dim E = k . The purpose of this paper is to study the structure of E when dim X - 2k -1 and dim E = k . The main result obtained in this paper is as follow:Let X be a smooth projective variety of dimensin 2k -1, assume thatf:X→Y is a small contraction and E is the exceptinal locus. If E is a k dimensional smooth subvariety and f(E) is a point, then E is a k dimensional...