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• Print publication year: 2016
• Online publication date: March 2016

# Chapter 1 - Introducing the Chow ring

## Summary

Keynote Questions

As we indicated in the introduction, we will preface each chapter of this book with a series of “keynote questions:” examples of the sort of concrete problems that can be solved using the ideas and techniques introduced in that chapter. In general, the answers to these questions will be found in the same chapter. In the present case, we will not develop our roster of examples sufficiently to answer the keynote questions below until the second chapter; we include them here so that the reader can have some idea of “what the subject is good for” in advance.

(1) Let F0, F1 and F2 ∈ k[X, Y, Z] be three general homogeneous cubic polynomials in three variables. Up to scalars, how many linear combinations t0F0+t1F1+t2F2 factor as a product of a linear and a quadratic polynomial? (Answer on page 65.)

(2) Let F0, F1, F2 and F3 ∈ k[X, Y, Z] be four general homogeneous cubic polynomials in three variables. How many linear combinations t0F0 + t1F1 + t2F2 + t3F3 factor as a product of three linear polynomials? (Answer on page 65.)

(3) If A, B, C are general homogeneous quadratic polynomials in three variables, for how many triples t = (t0, t1, t2) do we have

(A(t), B(t), C(t)) =. t0, t1, t2?