\II$. Hence, $x_h\in\II$ for some $h\in[1, i_{1}-1]$ or $1\in \II$. But $b_{i,h}<0$ implies that $h<i$ and we obtain the desired contradiction. The theorem is proved.

The theorem has the following corollary which was also independently proved by Muller very recently [@Mu; @1], though in more generality.

\[co:acyclic smooth\] Let $\AA$ be as in Theorem \[th:acyclic\]. Then, the variety $X$ defined by $\AA=\CC[X]$ is smooth.

The singular subset is contained in a Poisson ideal of co-dimension greater or equal to one by a result of Polishchuk ([@Pol]). It is well known that the Poisson ideal must be contained in a proper Poisson prime ideal (see also [@ZW; @tpc]). The assertion follows.

The assumption that the cluster algebra has even rank is very important. Indeed, Muller has recently shown that the variety corresponding to the cluster algebra of type $A_3$ has a singularity ([@Mu Section 6.2]).

Symplectic Structure
--------------------

### Symplectic geometry of Poisson varieties

In this section we will recall some well-known properties of the symplectic structure on Poisson varieties. Our discussion follows along the lines of [@brown-goodearl Part III.5]. If $A$ is a Poisson algebra over a field $k$, then each $a\in A$ defines a derivation $X_a$ on $A$ via 