Let $X$ be a smooth projective variety over an algebraically closed field $K$ of dimension greater than $1$. Suppose there exists a flat projective morphism $f:X \to \mathbb{P}^n$ for some $n \ge 1$. Let $W \subset X$ be a divisor in $X$ which is flat over $\mathbb{P}^n$. Under what condition on $f$ or $X$, does this imply that there exist a rational section to $f$?

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of dimension greater than $1$. Suppose there exists a flat projective morphism $f:X \to \mathbb{P}^n$ for some $n \ge 1$. Suppose there exists a divisor in $X$ which is flat over $\mathbb{P}^n$. Under what condition on $f$ or $X$, does this imply that there exist a rational section to $f$?

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of dimension greater than $1$. Suppose there exists a flat projective morphism $f:X \to \mathbb{P}^n$ for some $n \ge 1$. Let $W \subset X$ be a divisor in $X$ which is flat over $\mathbb{P}^n$. Under what condition on $f$ or $X$, does this imply that there exist a section to $f$?

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of dimension greater than $1$. Suppose there exists a flat projective morphism $f:X \to \mathbb{P}^n$ for some $n \ge 1$. Let $W \subset X$ be a divisor in $X$ which is flat over $\mathbb{P}^n$. Under what condition on $f$ or $X$, does this imply that there exist a rational section to $f$?