  In this paper we investigate the algebraic geometric nature of a solution of
the Yang-Baxter equation based on the quantum deformation of the centrally
extended $sl(2|2)$ superalgebra proposed by Beisert and Koroteev \cite{BEKO}.
We derive an alternative representation for the $\mathrm{R}$-matrix in which
the matrix elements are given in terms of rational functions depending on
weights sited on a degree six surface. For generic gauge the weights geometry
are governed by a genus one ruled surface while for a symmetric gauge choice
the weights lie instead on a genus five curve. We have written down the
polynomial identities satisfied by the $\mathrm{R}$-matrix entries needed to
uncover the corresponding geometric properties. For arbitrary gauge the
$\mathrm{R}$-matrix geometry is argued to be birational to the direct product
$\mathbb{CP}^1 \times \mathbb{CP}^1 \times \mathrm{A}$ where $\mathrm{A}$ is an
Abelian surface. For the symmetric gauge we present evidences that the
geometric content is that of a surface of general type lying on the so-called
Severi line with irregularity two and geometric genus nine. We discuss
potential geometric degenerations when the two free couplings are restricted to
certain one-dimensional subspaces.
