Rational equivalence
If you have taken linear algebra, then you are familiar with the notion of isomophism: two vector spaces can look different at first glance, but for all intents and purposes, share all relevant properties. There is a similar notion for algebraic varieties. Here is the idea.
Suppose that your variety is defined by a single polynomial:
It is a bit of a mess, but there is something simple going on. The polynomial is just a product of three factors. What if we changed variables and let
.
In terms of these, our polynomial takes on a much more appealing form:Strictly speaking, the varieties defined by these two polynomials are different, but are clearly related. If the change of variables is a polynomial, there are polynomially equivalent, if it is a rational function, rationally equivalent.
Project: Start by reading Chapter 5 book in investigating this problem. It has more than enough content for a wonderful project. Your goal is to understand the relationship between rationally equivalent varieties. How similar are they? And what are possible differences? Does rational equivalence help us to solve any problems we have talked about so far in a nicer way?