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We first prove that $Y$ is a klt Fano variety if ${\\rm deg} \\, f \\ge C$ for some constant $C = C(n, d)$ depending only on $n$ and $d$. Next we prove an optimal upper bound ${\\rm deg} \\, f \\le {\\rm deg} \\, X$ provided that $Y$ is factorial, ${\\rm deg} \\, f$ is prime and ${\\rm deg} \\, f \\ge E(n)$ for some constant $E(n)$ (with $E(n) = n(n+1)$ when $Y$ is smooth). 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