# compatible resolutions of singularities

Let $U$ and $V$ be complex vector spaces with an action of a finite group $G$. Denote by $P_{G,d}$ the space of $G$-equivariant polynomial maps $U\longrightarrow V$ with degree less than or equal to $d$.

There is an evaluation map $$ev_d:P_{G,d}\times U\longrightarrow V$$ Let $X_d\subset P_{G,d}$ be the variety $ev_d^{-1}(0)$ minus all irreducible components on which the action of $G$ is not effective. For $d$ large enough, $X_d$ is nonsingular at every point with trivial isotropy subgroup.

Each $X_d$ has a resolution $X'_d\longrightarrow X_d$ of its singularities. My question is this: Can I choose these resolutions compatibly for $d$ large enough so that there are inclusions $X'_d\longrightarrow X'_{d+1}$ for all $d$ large enough?

I would be happy enough if you could answer this question in the case when the action of $G$ is free on $U\setminus \{0\}$. In this case, the projection $P_{G,d+1}\longrightarrow P_{G,d}$ pulls back $X_d$ restricted to $0\in U$ to $X_{d+1}$ restricted to $0\in U$. In this case, I was hoping that the sequence of blowups of $P_{G,d}\times U$ required to desingularize $X_d$ could just be lifted using the projection $P_{G,d+1}\longrightarrow P_{G,d}$ to a sequence blowups of $P_{G,d+1}\times U$ which desingularize $X_{d+1}$.

(My background is symplectic topology and pseudo-holomorhpic curve theory, so please give your answers as low tech as possible. The answer would help to define integer curve counts in symplectic manifolds using a method outlined by Fukaya and Ono.)

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