  We construct a lattice cohomology ${\mathbb H}^*(C,o)=\oplus_{q\geq
0}{\mathbb H}^q(C,o)$ and a graded root ${\mathfrak R}(C,o)$ to any complex
isolated curve singularity $(C,o)$. Each ${\mathbb H}^q(C,o)$ is a ${\mathbb
Z}$-graded ${\mathbb Z}[U]$-module. The Euler characteristic of ${\mathbb
H}^*(C,o)$ is the delta-invariant of $(C,o)$. The construction is based on the
multivariable Hilbert series of the multifiltration provided by valuations of
the normalization.
  Several examples are discussed, e.g. Gorenstein curves (where an additional
symmetry is established), plane curves (in particular, Newton non-degenerate
ones), ordinary $r$-tuples.
  We also prove that a flat deformation $(C_t,o)_{t\in ({\mathbb C},0)}$ of
isolated curve singularities induces an explicit degree zero graded ${\mathbb
Z}[U]$-module morphism ${\mathbb H}^0(C_{t=0},o)\to {\mathbb
H}^0(C_{t\not=0},o)$, and a graded (graph) map of degree zero at the level of
graded roots ${\mathfrak R}(C_{t\not=0},o)\to{\mathfrak R}(C_{t=0},o)$.
  In the treatment of the deformation functor we need a second construction of
the lattice cohomology in terms of the system of linear subspace arrangements
associated with the above filtration.
