  Let $X$ be a smooth projective variety. Define a stable map $f:C\to X$ to be
"eventually smoothable" if there is an embedding $X\hookrightarrow\mathbb{P}^N$
such that $(C,f)$ occurs as the limit of a $1$-parameter family of stable maps
to $\mathbb{P}^N$ with smooth domain curves. Via an explicit
deformation-theoretic construction, we produce a large class of stable maps
(called "stable maps with model ghosts"), and show that they are eventually
smoothable.
