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Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ and } l_+-l_- \in L\}$. Does the ideal $J$ cutting $\overline{N^{\vee}_{X_L} \mathbb{C}^n}$ out of $T^* \mathbb{C^n}$ have a nice combinatorial description? I have an example which suggests that the ideal $J$ does not need to be binomial.

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