TITLE: Finite morphisms to projective space
QUESTION [8 upvotes]: 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 $n=1$.

REPLY [13 votes]: The answer is no in general. For an easy $2$-dimensional case, take for example $X=\mathbb{P}^1\times \mathbb{P}^1$, in which case $d=2$. Then, only the multiples of $d$ are possible for $d'$. Indeed, take a morphism $X\to \mathbb{P}^2$, and denote by $D$ the divisor of its linear system, which satisfies $D^2=d'$. The Picard group of $X$ is generated by $f_1,f_2$, the two fibres of the two projections, and $D=af_1+bf_2$, so $d'=D^2=2ab$.