Contact with lines

In a similar way we can argue for third order contacts with lines. Let X CP n be a smooth projective variety. On D (X) we consider the evaluation morphism... 

Proposition 3.2.19, Let X C P n be a smooth variety of dimension d. If the codimension of the locus, where X has third order contact with lines, has codimension 2N — 3d + 1, then its class is... 

As we know the Chow rings of X) and D (X), and the Chern classes of (H)2m and (H)f can be expressed in terms of the generators of these cohomology rings, we can in principle compute the classes of second order contacts with m-planes and the classes of third order contacts with lines. Note however that the Chow ring of D fn(X) is quite complicated for m > 2. For the explicit computation we will therefore restrict ourselves to the case of contacts with lines. We compute these classes for small N with the help of a computer. The total Segre class of ((H)2) is... 

Semple bundles and the formula for contacts with lines... 

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