Let $\pi:X \to S$ be an arithmetic surface, i.e. a flat, projective, scheme of relative dimension 1 over a Dedekind scheme $S$. So $X$ is a 2-dimensional excellent scheme. Suppose that the generic fiber $X_\eta/K$ is a smooth, geometrically connected curve of genus 0, and that $X$ is regular.
Since $\pi$ is a local complete intersection, the canonical sheaf $\omega_{X/S}$ on $X$ is an invertible sheaf. The dual $\omega_{X/S}^\vee$ restricts to a very ample sheaf on $X_\eta$, with global sections of rank 3.
Question 1: Is $\omega_{X/S}^\vee$ also very ample?
Now suppose that $Y$ is an arithmetic surface with very ample anticanonical bundle $\omega_{Y/S}^\vee$, and that $\rho:X \to Y$ is a birational morphism.
Question 2: Under what conditions is the anticanonical bundle $\omega_{X/S}^\vee$ also (very) ample?