Let $X \subseteq \mathbb{C}^n$ be a complex affine variety and $\tilde{X} \to X$ a surjective proper morphism where $\tilde{X}$ is smooth. Is it true that every morphism $\mathbb{C} \to X$ can be lifted to $\mathbb{C} \to \tilde{X}$?


I'm particularly interested in the case where $X$ has an isolated singularity and $\tilde{X} \to X$ is a resolution.



Lifting property for proper morphism
Regarding your general question, the answer is *no*.


Take any hyperbolic projective variety $Y$ (for instance, a ball quotient) of dimension $n$, and project it generically onto $\mathbb{P}^n$. Removing a hyperplane from $\mathbb{P}^n$ and its preimage from $Y$, we get a surjective finite morphism $f \colon Y^{\circ} \to \mathbb{C}^n$. There are plenty of morphisms $\mathbb{C} \to \mathbb{C}^n$, but none of them lifts to $Y^{\circ}$, since by the hyperbolicity assumption there are no non-constant entire curves in $Y^{\circ}$.


As a toy model, you can consider the case where $Y$ is a genus $2$ curve mapping $2:1$ to $\mathbb{P}^1$, and then remove a point from $\mathbb{P}^1$ and the two points over it from $Y$.


Regarding your question about the isolated singularity, the answer is *yes*. In fact, the resolution is an isomorphism away from the singular point $p \in X$. Thus, you can take the image of your entire curve in $X-{p}$, lift it to the resolution via the resolution map and then consider the Zariski closure in $\tilde{X}$.


