Recall that a variety $X$ over a finite field $k$ is said to be pure if the eigenvalues of the Frobenius on $i^{\mathrm{th}}$ etale cohomology of $\overline{X}:=X\otimes_k \overline{k}$ have absolutely value $q^{\frac{i}{2}}$.

Part III of the Weil conjectures (proved by Deligne) states that every smooth projective variety is pure.

Question: What are some examples of varieties which are neither smooth nor projective, but that are nonetheless pure?

It would be helpful if a brief explanation is given regarding how one proves purity in these examples (and/or a reference).

Recall that a variety $X$ over a finite field $k$ is said to be pure if the eigenvalues of the Frobenius on $i^{\mathrm{th}}$ etale cohomology of $\overline{X}:=X\otimes_k \overline{k}$ have absolutely value $q^{\frac{i}{2}}$.

Part III of the Weil conjectures (proved by Deligne) states that every smooth projective variety is pure.

There are, however, interesting examples of pure varieties which are not smooth projective. For instance, Springer fibres are singular projective and pure. Similarly, quiver varieties are smooth not projective but pure.

Question: What are some examples of varieties which are neither smooth nor projective, but that are nonetheless pure?

It would be helpful if a brief explanation is given regarding how one proves purity in these examples (and/or a reference).