definition of change of variable in Silverman

Question

Silverman mentions in his book The Arithmetic of Elliptic Curves the notion of a 'change of variable', without formally defining it. He did formally define what it means to have a rational map between (projective) varieties, and when such a rational map is a morphism. My guess is that a change of variable is the same thing as a birational map between two varieties (or maybe solely in the case of (not necessarily smooth) curves), but I'm not sure. Could someone confirm? Thanks.

Answer 1

A change of variables in this context is a map $[X_0:...:X_n] \to [X_0':...:X_n']$ given by a matrix $A \in GL_{n+1}(K)$ (in fact in $PGL_{n+1}(K)$) acting in the obvious way. Put another way, it is an automorphism of the underlying projective space $\mathbb{P}^n$ - hence the name "change of coordinates", suggested in a way by some new choice of "basis for the vector space" or "coordinate directions".