Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p\in X$. 


If $\dim X>0$, then does there necessarily exist a smooth curve $C$ on $X$ such that $f(p)$ is a smooth point of the closure of $f(C)$ in $Y$ ?



Image of smooth curve containing the image of a point as smooth point
That is not true. Let $k$ be an algebraically closed field of characteristic prime to $34$. Let the domain of the morphism be a Zariski open neighborhood of the origin in $\mathbb{A}^2\_k$ with coordinates $(s,t)$. Let the target of the morphism be a Zariski open neighborhood of the origin in $\mathbb{A}^2$ with coordinates $(u,v)$. Consider the following quasi-finite morphism, $$f:\mathbb{A}^2\to \mathbb{A}^2, \ \ f^\*u = s^3 + t^5, \ f^\*v = t^7 + s^{11}.$$ 


**Finiteness on appropriate neighborhoods of the origin.** By Bezout's Theorem, the generic degree of this morphism is $5\cdot 11 = 55$. The fiber over the origin is a complete intersection zero-cycle in the affine plane. The length of the component supported at the origin is $3\times 7 = 21$. Since $f^\*u$ equals zero on this zero-cycle, the residual zero-cycle has coordinates $(s,t) = (a^5,-a^3)$ for some $a\in k^\times$. Plugging into $f^\*v$, also $a^{34}-1 = 0$. This has $34$ distinct solutions, and thus the residual has length $34$. Altogether, the zero-cycle has length $55$. Thus, this quasi-finite, flat morphism is finite when restricted to an appropriate Zariski open neighborhood of the origin in the target and inverse image Zariski open neighborhood of the origin in the domain.


**Reduction to power series computations.** Consider the origin, $p$, in the domain. Every smooth arc of a curve at $C$ is either (Case I) the graph of a function $t=q(s)$ for $q\in sk[[s]]$, or (Case II) the graph of a function $s=r(t)$ for $r\in tk[[t]].$


**Case I.** The case $q(s)=0$ gives an arc $f(s) = (s^3,s^{11})$, which is a hypercusp. Thus, $q(s)$ is not identically zero. Therefore $q(s)$ has a power series expansion, $$q(s) = cs^m(1+sw(s)), \ \ c\neq 0, \ m\geq 1, \ w(s)\in k[[s]].$$ This gives, $$f^\*u(s,q(s)) = s^3(1+c^5s^{5m-3}(1+sw(s))^5), \ \ f^\*v(s,q(s)) = c^7s^{7m}(1+sw(s))^7 + s^{11}.$$ If this is the composition of a ramified morphism of smooth curves, $$\phi:C\to D,$$ followed by an immersion, $$\iota:D\hookrightarrow \mathbb{A}^2,$$ then $\iota^\*u$ must be a local coordinate on $D$. Thus, $\iota^\*v$ must be a power series in $\iota^\*u$, say $$\iota^\*v = b(\iota^\*u)^n + ... , \ \ b\neq 0.$$ Pulling back this identity by $\phi$ gives the identity, $$c^7s^{7m}(1+sw(s))^7 + s^{11} = bs^{3n}(1+c^5s^{5m-3}(1+sw(s))^5)^n + ...$$ If $m$ equals $1$, the lowest power of $s$ with nonzero coefficient on the left-hand side is $7$, and this is not a multiple of $3$. If $m\geq 2$, the lowest power is $11$, and this is also not a multiple of $3$. Thus, Case I leads to a contradiction.


**Case II.** The case $r(t)=0$ gives an arc $f(t) = (t^5,t^{7})$, which is a hypercusp. Thus, $r(t)$ is not identically zero. Therefore $r(t)$ has a power series expansion, $$r(t) = ct^m(1+tw(t)), \ \ w(t)\in k[[s]].$$ If $m$ equals $1$, then $r(t)$ is a formal isomorphism, and we can find an inverse function $t=q(s)$. By Case I, this leads to a contradiction. Therefore, $m$ is at least $2$. 


This gives, $$f^\*u(r(t),t) = t^5(1+c^3s^{3m-5}(1+tw(t))^3), \ \ f^\*v(r(t),t) = t^7 + c^{11}t^{11m}(1+tw(t))^{11}.$$ If this is the composition of a ramified morphism of smooth curves, $$\phi:C\to D,$$ followed by an immersion, $$\iota:D\hookrightarrow \mathbb{A}^2,$$ then $\iota^\*u$ must be a local coordinate on $D$. Thus, $\iota^\*v$ must be a power series in $\iota^\*u$, say $$\iota^\*v = b(\iota^\*u)^n + ... , \ \ b\neq 0.$$ Pulling back this identity by $\phi$ gives the identity, $$t^7 + c^{11}t^{11m}(1+tw(t))^{11} = bt^{5n}(1+c^3t^{3m-5}(1+tw(t))^3)^n + ...$$ The lowest power of $t$ on the left-hand side is $7$. Since $7$ is not a multiple of $5$, this is a contradiction in Case II.


Therefore, there is no smooth arc of a curve $C$ at $p$ such that the restriction of $f$ to $C$ factors as a ramified morphism of smooth curves followed by an immersion..


