When is the Hom-scheme connected? Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor $\mathrm{Hom}_K(\mathrm{Spec}(A),\mathrm{Spec}(B))$ from $(\mathbf{Sch}/K)$ to $(\mathbf{Sets})$, defined by 
$$ \mathrm{Hom}_K(\mathrm{Spec}(A),\mathrm{Spec}(B))(T):=\mathrm{Hom}_T(\mathrm{Spec}(A)\times_K T, \mathrm{Spec}(B)\times_K T),$$
is represented by an open subscheme $U$ of $\mathrm{Hilb}(\mathrm{Spec}(A)\times_K \mathrm{Spec}(B)/K)$.
Do we know when $U$ is connected? Thanks!
 A: I only just noticed this question.  I agree with Andrew Stout, but I am afraid that I disagree with Dmitry Vaintrob.  To make Dmitry's example precise, assume $B$ is a graded, Artinian (commutative, unital) $K$-algebra with residue field $K$, i.e., $$B = B_0 \oplus \dots \oplus B_q \oplus \dots \oplus B_n,$$ where $B_0 = K$.  This is the case, for instance, if we take $\text{Spec}(B)$ to be the infinitesimal neighborhood of the vertex in an affine cone with nilpotency index $n$.
There is a distinguished graded $K$-algebra homomorphism, $$ z : B \to A,\ \ z(b_0) = b_0,\ \ z(b_q) = 0,\  q > 0.$$  Moreover, there is a graded $K$-algebra homomorphism, $$ \mu^*: B[T] \to B[T], \ \ \mu^*(b_qT^d) = b_qT^{d+q}.$$  For any $K$-algebra homomorphism, $w:B\to A$, define the associated $K$-algebra homomorphism $w\otimes \text{Id}_{K[T]}$ as follows, $$ w\otimes \text{Id}_{K[T]} : B[T] \to A[T], \ \ (w\otimes\text{Id}_{K[T]})(bT^d) = w(b)T^d. $$  Now consider the composite $K$-algebra homomorphism, $$\widetilde{w}:B[T]\to A[T], \ \ \widetilde{w} = (w\otimes \text{Id}_{K[T]})\circ \mu^*.$$  When we specialize $\widetilde{w}$ via evaluation at $T=1$, then we recover $w$.  However, when we specialize via evaluation at $T=0$, we get the "vertex" $z$.  Interpreted differently, $\widetilde{w}$ gives a $K$-morphism, $$\widetilde{w}:\mathbb{A}^1_K \to \text{Hom}_K(\text{Spec}(A),\text{Spec}(B)),$$ that sends $1$ to $w$ and sends $0$ to the vertex $z$.  Thus the Hom scheme is representable by a connected $K$-scheme when $B$ is graded and Artinian.
