The varieties of second order data

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 D2m(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 Oy,x/tn3x z of Ox,x- Assume for the moment that the ground field is C and x Y C X, is a 

Notation. In chapter 3 and 4 we will often use the Grassmannian bundle associated to a vector bundle. So we fix some notations for these. 

Notation. For subschemes Zi, Z2 of a scheme S with ideal sheaves lzlf lz2 respectively in Os, let Z Z2 denote the subscheme Z of S whose ideal sheaf Tz is given by lz = 1Zl Iz2- 

As above we will write Z C Z2, to mean that Z is a subscheme of Z2- In this case we will write IZl/Z2 for the ideal of Zx in Z2. 

Definition 3.1.1. Let U (X) be the contravariant functor from the category of noetherian k-schemes to the category of sets which for noetherian fc-schemes S, T and a morphism / S — T is given by  

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In [Le Barz (10)] these varieties are shown to be smooth for X a smooth variety over C. The Ei X) are irreducible divisors in H3(X). D (X) is the variety of second order data on X, which we want to study in more detail in chapter 3. 

Later we will see that D X) is reduced and even smooth. D2n(X) is called the variety of second order data of m-dimensional subvarieties of X. Analogously we define D n(X) as the closed subscheme of X x Xtm+1l that represents the functor given by... 

For a surface 5 the variety D2(S) is considered in the literature (using a slightly different definition). It is called the variety of second order data on S and denoted... 

D iX) as the variety of second order data of m-dimensional subvarieties of X. 

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