I'm trying to learn about deformation theory.
Consider $k[x,y]/\left< y^2 - x^2\right>$.
To deform $k[x,y]/\left< y^2 - x^2\right>$ to make it look like $k[x,y]/\left< y^2\right>$, one would introduce a parameter $k[x,y,t]/\left< y^2 - t x^2\right>$ and make $t\rightarrow 0$.
Here's my question: if $x$, $y$ and $t$ each has weight 1, then $y^2 - t x^2$ is no longer homogeneous. When doing deformation theory, does the polynomial of interest need to be homogeneous?
In order to make $y^2 - t x^2$ homogeneous, would it be okay if I impose the following weights?
My first choice is to impose $wt(x)=wt(y)=1$ while $wt(t)=0$, but this isn't correct, right, since then $t$ is thought of as a constant? What goes wrong here?
My next choice is to impose $wt(y)=3$, $wt(x)=2$, and $wt(t)=2$.
Thanks for your time.