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Sherman-Zelevinsky and Cerulli constructed canonically positive bases incluster algebras associated to affine quivers having at most three vertices.Both constructions involve cluster monomials and normalized Chebyshevpolynomials of the first kind evaluated at a certain "imaginary" element in thecluster algebra. Using this combinatorial description, it is possible to definefor any affine quiver $Q$ a set $\mathcal B(Q)$ which is conjectured to be thecanonically positive basis of the acyclic cluster algebra $\mathcal A(Q)$.  In this article, we provide a geometric realization of the elements in$\mathcal B(Q)$ in terms of the representation theory of $Q$. This is done byintroducing an analogue of the Caldero-Chapoton cluster character where theusual quiver Grassmannian is replaced by a constructible subset calledtransverse quiver Grassmannian.

Transverse Quiver Grassmannians and Bases in Affine Cluster Algebras