General Results on the Hilbert scheme

we recall the definition of the Hilbert scheme in general (not necessarily of points, nor on a surface). Let X be a projective scheme over an algebraically closed field k and Gx(l) an ample line bundle on X. We consider the contra,variant functor Hilbx from the 

Namely, Hilbx is a functor which associates a scheme U with a set of families of closed subschemes in X parameterized by U. Let tt Z f/ be the projection. For u E U, the Hilbert polynomial in u is defined by 

Theorem 1.1 (Grothendieck [34]). The functor Hilbx is representable by a projective scheme Hilb.  

This means that there exists a universal family Z on Hilb, and that every family on U is induced by some morphism 0 17 —s- Hilb - 

The proof of this theorem is not given in this note. But we shall give a concrete description when P is a constant polynomial and X is an affine plane A. The reader who has no interests on the proof should accept the above theorem, and proceed. We do not need the proof of the theorem in this note. 

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Within any decoupling scheme there are only a few restrictions on the choice of the transformations U. First, they have to be unitary and analytic (holomorphic) functions on a suitable domain of the one-electron Hilbert space V, since any parametrization has necessarily to be expanded in a Taylor series around W = 0 for the sake of comparability but also for later application in nested decoupling procedures (see chapter 12). Second, they have to permit a decomposition of in even terms of well-defined order in a given expansion parameter of the Hamiltonian (such as 1/c or V). It is thus possible to parametrize U without loss of generality by a power-series ansatz in terms of an antihermitean operator W, where unitarity of the resulting power series is the only constraint. In the next section this most general parametrization of U is discussed. 

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