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Suppose $X$ is a regular surface, i.e. $H^1(X)=0$, e.g. a toric surface. Then $Y$ is a smooth rational curve. So $X$ has a very ample smooth rational curve. This immediately requires $X$ to be rational by taking a pencil of smooth rational curves. Does every rational surface have such a curve? It's ok for projective space (lines) and Hirzebruch surfaces (high degree sections).