The varieties of second and higher order data

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.  

In section 3.3 we introduce the Semple bundle varieties Fn(X), which parametrize higher order data of curves on X in a slightly different sense. We use them to show a general formula for the number of higher order contacts of a smooth projective variety X C Pjv with lines in P/v. 

Arrondo, Sols and Speiser [Arrondo-Sols-Speiser (1)] have independently constructed new contact varieties for m-dimensional subvarieties of a given variety X, for which they also give a number of applications. Their approach is different from the one of sections 3.1 and 3.2 and is in fact a generalization of the Semple bundle construction. 

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