Varieties of second order data

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. 

In section 3.2 we consider the varieties of higher order data D X). Their definition is a generalisation of that of D X). We show that only the varieties of third order data of curves and hypersurfaces are well-behaved, i.e. they are locally trivial bundles over the corresponding varieties of second order data with fibre a projective space. In particular D X) is a natural desingularisation of. Then we compute the Chow ring of these varieties. As an enumerative application of the results of chapter 3 we determine formulas for the numbers of second and third order contacts of a smooth projective variety X C Pn with linear subspaces of P. ... 

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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