Schemes with no finite morphisms onto themselves Let $X$ be an integral Noetherian scheme that has no automorphisms other than the identity. Is it possible to characterize (in a not completely tautological way) $X$ such that


*

*there is no finite morphism $X\rightarrow X$ surjective on the underlying topological spaces (other than the identity)?

*there is no finite flat morphism $X\rightarrow X$ surjective on the underlying topological spaces (other than the identity)? 


If $X$ is regular, then there is no distinction between two bullet points. 
Among spectra of fields, one can find examples satisfying the first (eq. second) bullet point ($\mathrm{Spec}\,\mathbb{Q}$) or not satisfying the first (eq. second) bullet point ($\mathrm{Spec}\,k$ for $k$ rigid non-perfect). 
 A: The question as stated seems to me too broad. Nevertheless, in some cases it is actually possible to provide a complete characterization. For instance, ruled surfaces admitting non-trivial, surjective endomorphisms are classified in
N. Nakayama: Ruled surfaces with non-trivial surjective endomorphisms, Kyushu J. Math. 56, No. 2, 433-446 (2002). ZBL1049.14029. 
A: A related question studied quite extensively in the literature is the existence of polarized endomorphisms. Let $X$ be a variety over $k$, for simplicity smooth and projective. An endomorphism $f\colon X\to X$ is polarized if there exists an ample line bundle $L$ on $X$ an an integer $n>1$ such that $f^* L\simeq L^n$. There is a conjecture (I think due to Fakhruddin) that if $k=\mathbb{C}$ and $X$ admits a polarized endomorphism, then up to a finite etale cover, $X$ is a bundle of toric varieties over an abelian variety. I believe the same should hold in positive characteristic under a separability assumption (the Frobenius is polarized endomorphism). 
