Fix an algebraically closed field $k$ (arbitrary characteristic), all schemes will be of finite type over $k$.

(Property *): I'm interested in (classes of) examples of schemes $X$ (irreducible, of dimension $n$) so that any morphism of schemes $\phi: X \rightarrow Y$ with $\dim Y < n$ is constant.

There are two examples I know of. Projective spaces $\mathbb{P}^n_k$ have this property and simple abelian varieties do too. (One may also put arbitrary non-reduced structures on these, see below).

$\textbf{Claim}$: More generally, if $X$ is a proper irreducible scheme so that every effective divisor is ample (so proper=projective), then $X$ has property (*). (Projective spaces have this property by definition, and simple abelian varieties do too by a general result from Mumford's book.)

Proof: (Eisenbud-Harris give a similar argument for the case of projective space). Let $X$ be as in the theorem and $\phi: X \rightarrow Y$ be a morphism with $Y$ of smaller dimension than $n = \dim X$. Without loss of generality, we may assume $\phi$ is surjective (hence we can pullback cartier divisors). Choose an effective Cartier divisor $D$ and a point $p \not \in |D|$ the support of $D$, but in the image of $\phi$ (since $\phi$ surjective). The pullback of an effective Cartier divisor is also effective, Cartier - hence ample. The pullback of the point will contain a complete curve (hence these two subschemes of $X$ meet by the Nakai-Moishezon criteria - contradicting that $p \not \in |D|$. $\square$

** Easy observations **

(1) Having property $*$ is not stable under blowing-up (Blow up of $\mathbb{P}^2$ is $\mathbb{P}^1 \times \mathbb{P}^1$.)

(2) If a scheme $X$ satisfies the claim, then by definition, so does $X_{red}$. Further, any thickening of $X$ has property $*$.

$\textbf{Proof: }$ To check that an $X$ satisfying the claim satisfies $*$, we did calculations in the intersection ring. This is invariant under changing the non-reduced structure. $\square$

** Questions **

Main question: Are there other (families of?) examples of schemes satisfying (*)?

(1) Does every scheme (no finiteness conditions!) have a dense affine open subset? This came up when I was thinking about this, and I realized I can't prove it offhand. Certainly it is true for irreducible schemes, and suffices to show it for connected schemes.

(2) Do you suspect that the only examples also satisfy the claim above? That is, have every effective divisor ample?

(3) Certainly all examples of schemes satisfying $*$ must be connected. Are there connected, but not irreducible examples?

I thought this was a little interesting (and admittidenly, I have no applications in mind.)

somefiniteness conditions. For example, if $X$ is the disjoint union of countably many copies of Spec $k$ (for some field $k$), then any affine subscheme will be finite, and so $X$ has no dense affine open subscheme. (This is related to the fact that an affine open is quasi-compact, so its closure will tend to want to be quasi-compact too, although I'm not sure in what generality that will literally be true.) Regards, $\endgroup$ – Emerton Dec 15 '12 at 0:44