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This page should have explicit examples of computations of Segre classes for various classes of cones. This should include standard examples such as vector bundles, but also cones which are not vector bundles. The cones page has good examples of this.
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http://ac.els-cdn.com/S0747717102000895/1-s2.0-S0747717102000895-main.pdf?_tid=59236850-8dcc-11e7-87b6-00000aacb35f&acdnat=1504129626_b93cb61ef06bfe90496445ac20e4a127
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