The fibres of a morphism

Let / X - Y be a morphism of varieties. The purpose of this section is to study the family of closed subsets of X consisting of the sets / 1(y), y EY. 

This reduces the study of the fibres of an arbitrary morphism to the case of dominating morphisms. Note also that a finite morphism is dominating if and only if it is surjective. 

Theorem 3. Let f X - Y be a dominating morphism of varieties and let r = dimJA — dimF. Then there exists a nonempty open set U C Y such that  

As in Theorem 2, we may as well replace Y by a nonempty open affine subset therefore, assume that Y is affine. Moreover, we can also reduce the proof easily to the case where X is affine. In fact, cover X by affine open sets A and let fo Xi - Y by the restriction of /. Let Ui C Y satisfy i) and ii) of the theorem for /. Let U = HUi. Then with this U, i) and ii) are correct for / itself. 

Now assume X and Y are affine, and let R and S be their coordinate rings. / defines a homomorphism 

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Proof. The kernel of a morphism of group schemes is representable. Consider the morphism of S-schemes G Ker( ). For any closed point s e S the morphism Gg Ker(V )s is flat as it is a surjection of group schemes over a field. Hence we can apply the criterion of flatness per fibre (EGA IV 11.3.11) to see that the morphism G -+ Ker( ) is flat and that Ker( ) is flat over S, Thus ip is admissible and as Im(< ) = Ker( ) follows from the above, the sequence is exact and (p is admissible also, ... 

For the classical geometers (pre-theory of schemes) it was natural to consider Q, as a well defined family of curves of P3. But flatness is a more natural condition If one wants to define a family not only as a set of varieties parametrized by the points of another variety, but as the set of ftbres of a morphism. We may try to construct a morphism whose fibres are the Cg s considering the closed subscheme % c P3xA1 defined by the ideal... 

The best intuitive description of when a morphism / X — Y of finite type is flat is that this is the case when the fibres of /, looked at locally near any point x X, form a continuously varying family of schemes. Suppose, for example, X and Y are varieties. Then, by D, if / is flat, X dominates Y. We saw in Ch. I, 8, that there is an open set U C Y such that all components of all fibres over points of U are n-dimensional, where n = dim A — dim Y. On the other hand, fibres over other points of Y may have dimension > n. This increase in dimension is clearly a big discontinuity of fibre type, and it can be shown that if / is flat, all components of all fibres of / have dimension n. In the other direction, for any morphism / X Y of varieties one would expect that almost all the fibres do form a continuous family, and indeed it can be shown that there always is some non-empty open U C Y such that res(/) f l U) —> U is flat. 

Definition 3.4.2 The sympleetic structure (Myw) is called integrable in the weak sense if there exists a surjective morphism / M — F of normal complex spaces whose general fibre is a completely isotropic connected submanifold in M. The sympleetic structure is called integrable without degeneracy if the base Y of the morphism f vs complex manifold and all the fibres of / are n-dimensional connected submanifolds in M,... 

The morphism Ag4 — -Spec(Z) is a locally complete intersection morphism of relative dimension g g + 1). This follows from a theorem of Mumford ([Oo] Theorem 2.3.3) since we know that every geometric fibre of Ag4 -> Spec(Z) has pure dimension g g +1) (see [NO] Theorem... 

Suppose we have an object (5, A, A) of A9ld Locally on S the group scheme Ker(A) is a pullback of the group scheme N by morphisms S — W. It is clear from the claim above that the fibre product S X gd Ag,s is in this case given by S Xw This proves that the morphism Agts Agtd is representable. It is separated and of finite type since this is true of the map i Z — W. This proves Proposition 1.5. E... 

A morphism f X — S can always be considered as a family of schemes if we view it as the totality of its fibres, if f is flat we call it a fiat family of schemes parametrized bv 5 X is called the total space of the family f If 5 is connected and s e 5 is a k-rational point, f is also called a flat family of deformations of the fibre X(s). When S=Spec(A) with A a local k-algebra with residue field k, the flat morphism f is called a local family of deformations of %(q), the ribre over the closed point 5 if moreover the local k-algebra A is artinian, then f is called an infinitesimal family of deformations, or an infinitesimal deformation, of X(o). 

XV ) If f X —> 5 is a flat and surjective morphism of algebraic and irreducible schemes, all fibres of f have pure dimension equal to d rn(X)-dim(5) This number is called the relative dimension of f... 

The assertion is true on a non-empty open set UcS. If U 5 we can find an irreducible and nonsingular curve T, a closed point t0cT and a morphism g- T 5 such that g(t0) t 5 U and g(TMt0 ) c U By pulling back we obtain a flat family f X > T in which the dimension of the fibres is not constant This implies that X is reducible and that one of its irreducible components does not dominate 5 This contradicts (XV)... 

XXIX) Let f % — S be a flat morphism of irreducible algebraic schemes. If s e 5 Is a point whose fibre (s) is smooth over 5pec(k(s)), j there is an open subset U of S containing s such that the restriction of... 

In 1 we (re)define the locally closed substacks Ag s C Ag d whose closures are the components of Ag d- We prove that all geometric fibres of the morphisms Ag s Spec(Z) are irreducible this answers a question of Mumford ([Mu], top of page 458 [GIT], page 189). It follows that the closures of 0 Fp are the irreducible components of 0 Fp. 

Lemma 1. Let f X -y Y be a finite morphism of noetherian schemes. For some y Y, assume that the fibre /-1(y) consists of one reduced point, with sheaf k (y) on it. Then there is an open neighbourhood U C Y of y such that... 

Actually it is no harder to prove a stronger theorem stating an equivalent fact for morphisms with fibres of positive dimension. This result involves the natural generalization of etale ... 

Proof. First assume / is smooth. It is clear that the definition of smoothness is such that whenever / X -> Y is smooth, all morphisms f obtained by a fibre product ... 

Proof. — This is a formal consequence of the fact that the morphism UIIV — X is an epimorphism of sheaves, the fact that U —> X is a monomorphism and that the Nisnevich sheaf associated to the fibre product U Xx V is indeed the fibre product in the category of sheaves. 

In the present subsection, the manifold M is assumed to be compact. Definition 3.4.1 The symplectic structure (Af,o ) is called completely inte-grable if there exists a proper surjective morphism of smooth complex manifolds f M N whose general fibre is a disconnected union of several completely isotropic n-dimensional tori (in particular, a general fibre may appear to be a torus). 

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