By Dan Laksov

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**Example text**

For each positive integer d we define a polynomial dt in Q[t] by t t(t − 1)(t − 2) · · · (t − d + 1) = td /d! + cd−1 td−1 + · · · + c0 = d d! and we let t 0 = 1. 3) Note. For each positive integer e we define an operator ∆e on all functions f : Z → Z by ∆e f (m) = f (m + e) − f (m). t We let ∆ = ∆1 . Then ∆ dt = d−1 . For each non–negative integer d we have that we have that ∆e t d = t+e t − d d =e t d defines a function Z → Z and td−1 + bd−2 td−2 + · · · + b0 . (d − 1)! Thus the polynomials ∆e 1t , ∆e 2t , .

Reordering the first s − r rows and coluns, if necessary, and using row and column operations, we can make the upper left (s − r) × (s − r)–matrix in the upper left corner the unit matrix. We can then use row and column operations on β to put β in a form where the r × (s − r)–matrix in the lower left corner and the (s − r) × (t − s + r)–matrix in the upper right corner are zero. Since we have assumed that Fr−1 (M )AP ⊆ P AP we have that the coordinates of the r × (t − s + r)–matrix in the lower right corner are in P AP .

1) Setup. Given a scheme S and a contravariant functor F from schemes over S to sets. All schemes and morphisms will be taken over S. Given a scheme X over S, we denote by hX the contravariant functor from schemes over S to sets which sends a scheme T to the set of S–homomorphisms hX (T ) = HomS (T, X) from T to X, and to a morphism h: U → T associates the map hX (h): hX (T ) → hX (U ) given by hX (h)(g) = gh, for all morphisms g: T → X. 2) Note. There is a natural bijection between elements in F (X) and morphisms of functors H: hX → F .

### Hilbert schemes by Dan Laksov

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Categories: Algebraic Geometry