Algebraically closed field

Let k be a (not necessarily algebraically closed) field and X a smooth quasipro-jective variety of dimension d over k. In this section we study the structure of the stratum (Xj j)re(j which parametrizes subschemes of X which are concentrated in a (variable) point in X. 

Let k be an algebraically closed field. In this section we review the methods of [Ellingsrud-Str0mme(l)] for the determination of a cell decomposition of and modify them in order get a cell decomposition and thus (for k = C) the homology of the strata Zt and Gt of Hilb"(fc[[x, y]]). Let R = fc[[ ,y]]. Let Hilb (A2,0) be the closed subscheme with the induced reduced structure of (A2) parametrizing subschemes with support 0. By lemma 2.1.4 we have... 

Theorem 2.2.3. [Bialynicki-Birula (1),(2)] Let X be a smooth projective variety over an algebraically closed field k with an action of Gm. Assume that the set of... 

Let X be a smooth projective variety of dimension d over an algebraically closed field k. In this section we want to define a variety D (X) of second order data of m-dimensional subvarieties of X for any non-negative integer m < d. A general point of D ln X) will correspond to the second order datum of the germ of a smooth m-dimensional subvariety Y C X in a point x X, i.e. to the quotient of Ox,x- Assume for the moment that the ground field is C and x Y C X, X is a smooth complex d-manifold and we have local coordinates zi,..., at x. Then Y is given by equations... 

Let X be a projective scheme over an algebraically closed field k. In [Beltrametti-Sommese (1)] the following definition was made ... 

Let X be a smooth projective variety over an algebraically closed field k. The easiest examples of zero-dimensional subschemes of X are the sets of n distinct points on X. These have of course length n, where the length of a zero-dimensional subscheme Z is dimkH°(Z,Oz)- On the other hand these points can also partially coincide and then the scheme structure becomes important. For instance subschemes of length 2 are either two distinct points or can be viewed as pairs (p, t), where p is a point of X and t is a tangent direction to X at p. 

First, 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... 

Suppose k is an algebraically closed field of characteristic p > 0 and suppose x Spec(fc) — Ag>5 is a geometric point given by the polarized abelian variety (X0, A0) over Spec (A ). In [Cr] the complete local ring of Agtd at x is computed. We will describe the result. 

Let (A, A) be a principally polarized abelian variety over an algebraically closed field k. If the characteristic of k is not equal to the prime p, then the kernel of multiplication by p on A(k) is a finite group isomorphic to (Z/pZ)29. The polarization A induces a nondegenerate alternating pairing A k)[p] x A( )[p] -+ pp(k). Hence, we can try to classify principally polarized abelian varieties with a symplectic basis for the group of points of order p. However, this no longer works in characteristic p. 

All rings will be commutative with 1. Once and for all we fix an algebraically closed field k All schemes will be assumed to be defined over k and noetherian and all sheaves will be quasi-coherent. By an algebraic scheme we mean a scheme of finite type over k. An integral scheme is one which is reduced and irreducible, if X and Y are schemes we will write X Y instead of XxkY < ). A variety is a reduced algebraic scheme a curve. resp. a surface, is a variety of pure dimension 1, resp. 2. 

XXIV) Let K be an algebraically closed field extension of k. A K-scheme of finite type X is nonsingular iff the structural morphism x - Spec(K) Is smooth. 

Lemma 9.5.1 Let C be an algebraically closed field of characteristic 0. Then,... 

If (L) = S and S has finite valency, the condition that 5 i(L) has exactly one element is related to a natural condition about the scheme ring of S over an algebraically closed field of characteristic 0. In fact, we shall see that S- (L)... 

Theorem 10.6.7 Assume that there exists an algebraically closed field C of characteristic 0 such that C L = C(L). Then S i(L) = 1. 

The main idea of the proof of Theorem 12.4.6 is to apply Theorem 9.1.7(ii) to an algebraically closed field C of characteristic 0. According to Lemma 9.2.5, the left hand side of the equation in Theorem 9.1.7(ii) is algebraic over the smallest unitary subring Z of C and the right hand side of that equation is in the smallest subfield of C. Thus, according to Lemma 8.2.5, both sides must be in Z, and one obtains an integral divisibility condition. 

Proposition 12.4.2 Let C be an algebraically closed field of characteristic 0, and let x be an irreducible character of CS of degree 2. Then there exists an element c in Cd such that the following hold. 

Theorem. Let G be an abelian affine group scheme over an algebraically closed field.. Then any irreducible representation of G is one-dimensional. 

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