$\gdef\CS{ {\C^*}}$ $\gdef\v{\vee}$ $\gdef\ycal{\mathcal{Y}}$ $\gdef\zcal{\mathcal{Z}}$

Consider a SES of lattices $$ L \to E \to N $$, where $E = \Z^N$. And consider its dual lattices $$ N^\v \to E^\v \to L^\v $$ If $A$ is a lattice, we let $A_\CS = A \otimes \CS$ to be the corresponding torus where $A$ is the cocharacter lattice.

Consider the basic case, where $$\zcal = \C^2_{x,y} \RM \{xy+1=0\}, \quad \mu: \ycal \to \CS, \mu(x,y)=xy+1. $$

In general, we have group-valued moment map being the composition $$ \mu_L: (\zcal)^N \xto{\mu_E} (\C^*)^N = E^\v_\CS \xto{Q_\CS} L^\v_\CS $$ and the affine hypertoric variety is obtained by algebraic symplectic reduction $$ \ycal_\beta = \mu_L^{-1}(\beta) / L_\CS, \quad \forall \beta \in L^\v_\CS. $$ For generic $\beta$, away from some hyperplane arrangements $\Delta_L$ in $L^\v_\CS$, we have $\ycal_\beta$ smooth affine.

Recall we have the moment map $\mu_L = Q \circ \mu_E$. We just define $$ Y_\beta = Q_\CS^{-1}(\beta). $$

We have map $$ \pi: \ycal_\beta \to Y_\beta $$ with singular loci at the divisor $D_\beta \In Y_\beta$.

There is no universal skeleton for all $\beta \in L^\v_\CS \RM \Delta_L$. For $\beta \in L^\v_{U(1)}$ and $\beta \in L^\v_{\R_+}$, we have some natural construction.