Let $f:X'\to X$ be a projective birational morphism between complete algebraic varieties. Assume that the exceptional locus ${\rm Exc}(f)$ is the support of an effective Cartier divisor, can we choose this divisor to be $f$-anti-ample?

When $X'$ is regular, $f$ is isomorphic to a blowup along a subscheme $Z$ such that ${\rm Exc}(f)=f^{-1}(Z)$ (GTM 52, II, E.x. 7.11(c)). As a consequence, the answer to the question above is affirmative when $X'$ is regular.

See Projectivity of blowups for a similar question and an affirmative answer when $X$ has only $\mathbb{Q}$-factorial singularities.

Let $f:X'\to X$ be a projective birational morphism between complete algebraic varieties. Assume that the exceptional locus ${\rm Exc}(f)$ is the support of an effective Cartier divisor, can we choose this divisor to be $f$-anti-ample?

When $X$ is regular, $f$ is isomorphic to a blowup along a subscheme $Z$ such that ${\rm Exc}(f)=f^{-1}(Z)$ (GTM 52, II, E.x. 7.11(c)). As a consequence, the answer to the question above is affirmative when $X$ is regular.

See Projectivity of blowups for a similar question and an affirmative answer when $X$ has only $\mathbb{Q}$-factorial singularities.