  We construct algebraic-geometric families of genus one (i.e. elliptic)
current and affine Lie algebras of Krichever-Novikov type. These families
deform the classical current, respectively affine Kac-Moody Lie algebras. The
construction is induced by the geometric process of degenerating the elliptic
curve to singular cubics. If the finite-dimensional Lie algebra defining the
infinite dimensional current algebra is simple then, even if restricted to
local families, the constructed families are non-equivalent to the trivial
family. In particular, we show that the current algebra is geometrically not
rigid, despite its formal rigidity. This shows that in the infinite-dimensional
Lie algebra case the relations between geometric deformations, formal
deformations and Lie algebra two-cohomology are not that close as in the
finite-dimensional case. The constructed families are e.g. of relevance in the
global operator approach to the Wess-Zumino-Witten-Novikov models appearing in
the quantization of Conformal Field Theory.
