Trivial bundle

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. ... 

Of course, all characters mentioned in this section coincide on trivial bundles, that is bundles (/ / 2. . Rn) such that Rt R2. . . = RH. Harmonic bundles however are not trivial in general on the contrary, they are complex enough to generate arbitrary orderings, even if those harmonic bundles are restricted to a particular kind. To illustrate this and also some other interesting relationships concerning harmonic bundles, we shall discuss in some detail the case of three alternatives. 

Finally, far from being trivial bundles, harmonic bundles are complex enough to represent any ordering. This is so even when harmonic bundles are restricted further to some particular kind, which then ensures uniqueness for the representation. This representation, which seems to be useful for analysing rationality, will be presented in full generality in the next section. For the moment we want to illustrate it in the simple case of three alternatives. As the reader will easily verify, the following list exhibits all the orderings possible on the set = x,y, jc) ... 

Statement (d) exhibits in precise form a peculiar feature of the Rawls aggregation device which we already observed earlier (see the example at the end of the last section). The selection problem for the Rawls-device does not vanish, except, of course, for trivial bundles. Or, to put it in the language of Social or Public Choice, the Rawls device depends crucially on interpersonal comparison (trivial cases excepted). (In fact, statements (c) and (d) are strongly related to questions raised in the theory of Social or Public Choice. Thus (c) provides a precise answer to the question of a purely ordinal behaviour of the utilitarian device, which is discussed e.g. by Mueller 1979, p. 176.)... 

The first, and simplest, example of a bundle is that of a direct product E = B X F, with the canonical projection being the projection onto the first term. Not surprisingly, this bundle is called a trivial bundle. 

The following are two distinct real line bundles (i.e., vector bundles whose fibers are R ) over S the trivial bundle and the twisted bundle, where the line changes orientation after passing once around the circle cf. Figure 8.1. 

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). 

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. 

Z f ( t1) s a l°ca y trivial fibre bundle with fibre Ar over... 

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. 

We now want to show that D (X) and D 1 X) are again Grassmannian bundles corresponding to vector bundles over D X) and D j l(X) respectively. Before doing this we want to show that these two cases are the only ones in which we can expect such a result (exept for the trivial case m — d). 

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

Consider the trivial line bundle V xC over V. Using x, we lift the G -action to U x C... 

In this picture, the correspondence between irreducible representations of F (except the trivial representation) and irreducible components of the exceptional set becomes concrete. It is realized by the tautological bundles V s. In [66, 5.8], we have shown the correspondence respects the multiplicative structures, one given by the tensor product and one given by the cup product. In fact, using (4.11), we can show that two matrices... 

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 relations termed (t.. . , E6 are all elementary in that they all have the smallest possible (non-trivial) number of indifference classes, namely 2. (There is exactly one ordering having 1 indifference class, namely the trivial order x — — z which can be also considered as elementary.) We leave it to the reader to verify that for every ordering R on A there exist uniquely determined elementary orderings Et and Ey such that (Eh Ef) is a harmonic bundle and R is the set-theoretical intersection of , and Ey. Thus, for example, jc > > z is the intersection of E and 2, ( 1, E2) being a harmonic bundle and there is no other bundle of elementary orderings having these two properties. 

The holomorphic symplectic structure on (C2) restricts to that on A. In particular, it implies that the canonical bundle is trivial, i.e. Kx = Ox- Hence the resolution is minimal. ... 

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