Again, considering the VGIT and LG problem. Read a bit BFK in the morning. The graded MF is really something.

Another thing that I realized is, at least for HMS of toric GIT, the choice of a cocharacter in $T_B$ corresponds to a 1PS in $T_A$. One probably should consider compactifying the base.

But I don't want to compactify, because I don't know what is the limit of that holomorphic function $W_t$.

When in doubt, do graph. I think that is a good approach. For example, consider the function on $(\C^*)^2$ $f(x,y) = x(1 + y + ty^2)$. As $t \to 0$, we know the term $t y^2$ will be important eventually when $y$ is large enough. So, we can include $t$ as part of the variables, and we have some terms in $\Z^3$. We are going to look at those 1PS, where $t$ variable has speed 1.

a la Webster. Let $G=\C^*$ acts on my space $X$ on $B$-side. Let $L$ be an $G$ equivariant ample line bundle. We consider the projective ring of $\oplus_n \Gamma(L^n)$, then taking $G$-invariant subring.