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No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.

(A paper claiming to give a negative answer to a well-known open problem had to be withdrawn substantially revised because the authors implicitly used the "fact" that the answer to this question is "Yes"!)

No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.

No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.

(A paper claiming to give a negative answer to a well-known open problem had to be withdrawn because the authors implicitly used the "fact" that the answer to this question is "Yes"!)

No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.

(A paper claiming to give a negative answer to a well-known open problem had to be withdrawn substantially revised because the authors implicitly used the "fact" that the answer to this question is "Yes"!)

No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.

(A paper claiming to give counterexamples to a well-known open problem had to be withdrawn because the authors implicitly used the "fact" that the answer to this question is "Yes"!)

No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.

(A paper claiming to give a negative answer to a well-known open problem had to be withdrawn because the authors implicitly used the "fact" that the answer to this question is "Yes"!)