Locally Trivial Bundles

In section 3.2 we consider the varieties of higher order data D X). Their definition is a generalisation of that of D X). We show that only the varieties of third order data of curves and hypersurfaces are well-behaved, i.e. they are locally trivial bundles over the corresponding varieties of second order data with fibre a projective space. In particular D X) is a natural desingularisation of. Then we compute the Chow ring of these varieties. As an enumerative application of the results of chapter 3 we determine formulas for the numbers of second and third order contacts of a smooth projective variety X C Pn with linear subspaces of P. ... 

The second chapter is devoted to computing the Betti numbers of Hilbert schemes of points. The main tool we want to use are the Weil conjectures. In section 2.1 we will study the structure of the closed subscheme of X which parametrizes subschemes of length nonl concentrated in a variable point of X. We will show that (X )rei is a locally trivial fibre bundle over X in the Zariski topology with fibre Hilbn( [[xi,... arj]]). We will then also globalize the stratification of Hilbn( [[xi,..., x ]]) from section 1.3 to a stratification of Some of the strata parametrize higher order data of smooth m-dimensional subvarieties Y C X for m < d. In chapter 3 we will study natural smooth compactifications of these strata. 

Lemma 2.1.4. n (X " j)red — X is a locally trivial fibre bundle in the Zariski topology with fibre Hilbn(J )red. 

Zt X) and Gt(X) are locally trivial fibre bundles over X with fibres Zt and Gt respectively. 

Then Dlm(X)o is via pi a locally trivial fibre bundle over Dlm 1 (W)0 with fibre ASd m)(, ) Phis is only a reformulation of remark 2.1.7. 

As Hilbn(P(i5)/X) is a locally trivial fibre bundle over X with fibre P, we see easily ... 

The second chapter is devoted to computing the Betti numbers of Hilbert schemes of points. The main tool we want to use are the Weil conjectures. In section 2.1 we will study the structure of the closed subscheme of which parametrizes subschemes of length n on X concentrated in a variable point of X. We will show that is a locally trivial fibre bundle over X in the Zariski topology... 

Then D m(X)0 is via pi a locally trivial fibre bundle over o with fibre... 

Example 2.3. — Let T be a Zariski torsor for a vector bundle over the smooth scheme X over S. Then the morphism T —> X is an A -weak equivalence. It follows from Lemma 2.16 applied to the class C of sheaves represented by smooth schemes over S which are affine (over Spec 2,)), Example 2.2 and the fact that any such torsor is trivial when the base is affine over Sped. More generaUy any smooth morphism Y X of schemes which is a locally trivial fibration in the Nisnevich topology with an A -contractible fiber is an A -weak equivalence. 

X SL fibre bundle with a fibre above is defined, which is locally trivial for all nonsero points of the disk The circle fibre (0,t),0 < t 1 is singular if a > 1. Now consider the manifold with points xi,... marked on it and encircle them with small disks. .., D. Then take the direct product P X and discard from it m full tori D X 5, 1 t m. In the place of these full tori glue in fibered full tori of the type (2.1). Denote the manifold obtained by P x5. It is a Seifert bundle and P is its base. 

In many situations the considered topological space has some additional associated structure that is locally trivial, but may contain some important information when considered globally. In such a case, the concept of a bundle may turn out to be useful. 

We shall consider only locally trivial bimdles hence we will just say fiber bimdle or sometimes just bundle. In some texts one allows a slightly more general type of bundles, where the fibers may be different over various path-connected components of B. We shall not make use of this generality. 

The base manifold of internal motion is a Riemannian manifold (Refs.70,71). This conformation space of a molecule has some good properties, but one conspicuously missing property is the general local structure. It is rather complicated and not as well-behaved as one might hope. The case of a triatomic is still simple. But it is to be noted that even though the base manifold here is a trivial bundle, the connection has nonvanishing curvature, i.e., it is not flat with respect to this connection (cf. also Refs.72-74). 

Locally in the etale topology on S we can choose a rigidified line bundle (i.e. a line bundle trivialized along the origin) L on X such that A(L) = A (see [GIT] Definition 6.2). This... 

The Born-Oppenheimer approximation, whose validity depends on there being a deep enough localized potential well in the electronic energy, has, however, been extensively treated. The mathematical approaches depend upon the theory of fiber bundles and the electronic Hamiltonian in these approaches is defined in terms of a fiber bundle. It is central to these approaches, however, that the fiber bundle should be trivial, that is that the base manifold and the basis for the fibers be describable as a direct product of Cartesian spaces. This is obviously possible with the decomposition choice made for O Eq. 2.42 but not obviously so in the choice made for O Eq. 2.43. 

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