(Connected) non-contractible schemes This question is motivated by this other question and this answer, which show that irreducible algebraic varieties and more generally integral schemes are contractible as topological spaces.
What are examples of connected non-contractible schemes? I expect some gluing involved in the answer. But could such a scheme also be affine?
 A: Take a union $X$ of a line $L$ and a conic $C$ in $\mathbb{A}^2$ (over some field) which intersect at two points $p$ and $q$.  Then I claim $X$ is not contractible, and in fact that $H_1(X)$ is nontrivial.  To prove this, cover $X$ by the open sets $U=X\setminus\{p\}$ and $V=X\setminus\{q\}$.  Note that $U$ and $V$ are path-connected, but $U\cap V$ is disconnected (its connected components are $L\setminus\{p,q\}$ and $C\setminus\{p,q\}$).  The Mayer-Vietoris sequence $$H_1(X)\to H_0(U\cap V)\to H_0(U)\oplus H_0(V)$$ then tells us that $H_1(X)$ must be nontrivial, since the map $H_0(U\cap V)\to H_0(U)\oplus H_0(V)$ is the same on every path-component of $U\cap V$ and thus is not injective.
The intuition here is that the space $X$ is like a circle, obtained by gluing together two contractible sets $L$ and $C$ which intersect at two points, just like a circle is obtained by gluing together two intervals that intersect at two points.  In fact, with a bit more work you can show that $X$ is weakly homotopy equivalent to a circle.
More generally, I suspect the following statement is true but have not carefully worked out a proof.  Let $X$ be a sober Noetherian topological space.  Let $P$ be the smallest collection of irreducible closed subsets of $X$ such that every irreducible component of $X$ is in $P$ and if $A,B\in P$, then every irreducible component of $A\cap B$ is in $P$.  We can consider $P$ as a poset with respect to inclusion.  Then $X$ is weakly homotopy equivalent to the nerve of $P$.
(Alternatively, you could consider the poset $Q$ which consists simply of all irreducible closed subsets of $X$, or equivalently the poset of points of $X$ with respect to the specialization order.  It should also be true that $X$ is weak equivalent to the nerve of $Q$.  This construction is somewhat more natural, but the poset $P$ above has the advantage of being finite and thus nice to compute with.)
