Some Extremal Contractions between Smooth Varieties Arising from Projective Geometry

Let ψ : X → Y be a proper morphism with connected fibers from a smooth projective variety X onto a normal variety Y , i.e. a contraction. If −KX is ψ-ample, then ψ is said to be an extremal contraction and if moreover Pic(X)/ψ(Pic(Y )) ≃ Z then ψ is said to be an elementary extremal contraction or a Fano-Mori contraction. Usually contractions are divided into two types: birational and of fiber type, i.e. with dim(X) > dim(Y ). The importance of this study and the parallel development of the notion of extremal ray and of the possible contractions of these rays was established by Mori in [18]. In dimension 2 and over an arbitrary field only three types of FanoMori contractions exist: the inverse of the blow-up of a reduced and geometrically irreducible set of points defined over the field (birational), conic bundle structure and del Pezzo surfaces (of fiber type) and this classification is essentially equivalent to the theory of minimal models in dimension 2. In dimension 3 the theory is more delicate but there is a complete description for algebraically closed fields given by Mori in characteristic zero and by Kollár for arbitrary algebraically closed fields, see for example [17], theorem 1.32 or [18]. The common feature of extremal elementary contractions in dimension 3 (and 2) is that if E ⊆ X is the exceptional locus of the contraction, i.e. the locus of positive dimensional fibers, then φ|E : E → φ(E) ⊆ Y is equidimensional. Moreover in the birational case (and over an algebraically closed field) these contractions are always the inverse of the blow-up of a (possible singular) point or of a smooth curve, see loc. cit. In dimension greater than three the situation is more complicated as it is shown by simple examples and no general result is known. Here we furnish some examples of Fano-Mori contractions ψ : X → Y between smooth projective varieties with dim(X) ≥ 4 defined over ”sufficiently” small fields and in arbitrary characteristic, having some exceptional fibers of unexpected dimension. All these contractions come from classical projective geometry, or more precisely they are associated to the projective geometry of remarkable classes of varieties: varieties with one apparent double, triple, quadruple point or varieties defining special Cremona transformations. Surely many examples of the phenomenon of non-equidimensionality of the fibers of φ|E were constructed and are well known at least in characteristic zero or