Semple bundles and the formula for contacts with lines

Semple bundles and the formula for contacts with lines 

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) the nth order data of curves on X. Like D (X) and D (X) they are smooth compactifications of c i y a tower of P i i-bundles over X (d = dim(X)). Remember that 

For simplicity we will assume during the whole of section 3.3 that the ground field is C. 

Definition 3.3.1. Let X be a smooth variety of dimension d. We define inductively varieties Fn(X) and vector bundles Gn(X) on Fn(X). Let 

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. 

Assume inductively that Fo(X). Fn-1(X) and Go(A ). Gn-i(X) are already defined. Assume furthermore that Gn-i(X) is a subbundle of the tangent bundle Tpn, (x) of rank d. Then we put 

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