I am reading a book and I am curious about a certain notion.
Consider $R = k[x_1,x_2,x_3,x_4,t]$ and let $G = \{\underbrace{x_1 x_3-x_2^2 + t x_3^2}_{f_1}, \underbrace{x_1 x_4-x_2 x_3 +t x_2^2}_{f_2} \}$.
When $t=0$, we obtain the standard twisted cubic.
After imposing the following weights to the variables: $ wt(x_i)=1 \mbox{ and } wt(t)=0. $ does this mean I am viewing $t$ as a constant, and would you say $f_1$ and $f_2$ are homogeneous of degree 2, rather than think of it as a mixed degree polynomial? $ $ What if I, instead, impose the weights to be $ wt(x_i)=2 \mbox{ and } wt(t)=1? $
What is the purpose of giving variables different weights?
Addendum: is there a geometric significance to the notion of weights?