Semple-bundle varieties

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. 

In this section we introduce the Semple bundle varieties Fn(X) of a smooth variety X. They parametrize in a slightly different sense than D (X j the nth order data of curves on X. Like D (X) and D (X) they are smooth compactifications of c by a tower of Pd i-bundles over X (d = dim(X)). Remember that... 

The Semple bundle varieties were first introduced in [Semple (1)]. In [Collino (1)], [Colley-Kennedy (1),(2)] they are considered for arbitrary smooth surfaces. The construction of Fn(X) for an arbitrary smooth projective variety X is an obvious generalisation. For our purposes it appears to be slightly more practical to use the tangent bundles instead of the cotangent bundles in the construction. 

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. 

Now we want to show that more generally the class in A (X) of the locus where a smooth projective variety X C P v has nth order contact with a given family of curves is properly interpreted a linear combination of monomials in H and the Chern classes of X. This linear combination will depend on the familyCr- We will not treat here the much more difficult question how to determine this linear combination for a given family Ct- The argument is similar to that at the end of section 3.2. First we will generalize the Semple bundles to a relative situation. 

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