Porteous formula

Now we want to compute the class of the complement D (X)<, = D (X) D (X)0. It parametrizes in a suitable sense the second order data of singular m-dimensional subvarieties of X. We will use a tool that will play a major role in the enumerative applications of higher order data in section 3.2, the Porteous formula. We will not quote the result in full generality but in the formulation in which we are going to use it. 

We again want to use the Porteous formula. Let H = OpN(l) be the hyperplane bundle on Pn- We will denote by the same letter its restriction to X and its first Chern class. 

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