Let $X$ be a projective variety of dimension n. Then there exists a finite surjective morphism $X \to \mathbf P^n$. Let $d$ be the minimal degree of such a finite surjective morphism.

Let $d^\prime > d$.

Question. Does there exist a finite surjective morphism $X\to \mathbf P^n$ of degree $d^\prime$?

The answer is yes for multiples of $d$, but what about the general case?

Also, by Riemann-Roch the answer is yes for $n=1$.