Hilbert scheme relative

Theorem. For every r and fi(t) as above the flag-Hilbert functor FHrB(t) is represented by a projective scheme FHr (t), called the flao-Hilbert scheme relative to >(t), and by a "universal family"... 

Where EM is the flag-HIlbert scheme relative to (t+1, 151-27), which parametrizes flags of the form... 

The Chow ring of relative Hilbert schemes of projective bundles... 

Hilbn(X/T) relative Hilbert scheme of subschemes of length n 2... 

Hilbn(P( E)/X ) relative Hilbert scheme of projective bundle 149... 

The last chapter is the most elementary and classical of the book. We describe the Chow ring of the relative Hilbert scheme of three points of a P2 bundle. The main example one has in mind is the tautological P2-bundle over the Grassmannian of two-planes in P". In this case it turns out hat our variety is a blow up of (Pn)t3l. This fact has been used in [Rossello (2)] to determine the Chow ring of (P3) 3). The techniques we use are mostly elementary, for instance a study of the relative Hilbert scheme of finite length subschemes in a P1-bundle I do however hope that the reader will find them useful in applications. 

In IG2) Grothendieck introduced the Hilbert scheme of lPr I prefer to associate one Hilbert scheme to each Hilbert polynomial p(t), so that in this volume the Hilbert scheme of JPr in the sense of [G2] is the disjoint union of all the Hilbert schemes of Pr relative to all different polynomials p(t)... 

We will prove that N is the Hilbert scheme of Pr relative to p(t) and that the universal family is the restriction of q to H. 

Hilbert scheme of closed subschemes of Pr containing Z relative to q(t), and by a universal family... 

From now on we will consider Hilbert schemes tiilbrp[Pg.141]

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