I know that maxSpec$\mathbb{C}[x,y]/(x^{2})$ is homeomorphic to the line $\mathbb{A}^{1}$. What can we say about Spec$\mathbb{C}[x, y]/(x(x-t))$ for fixed complex $t \neq 0$? That is, what does it look like? What happens as $t$ varies?
I suspect not much happens when $t$ is nonzero. When $t \rightarrow 0$ we get Spec$\mathbb{C}[x, y]/(x^{2})$, for which we know the maximum spectrum of above.
I know that $(x(x-t)) \subset (x)(x-t)$, and so Spec$\mathbb{C}[x, y]/(x(x-t)) \supset$ Spec$\mathbb{C}[x, y]/(x)$ $\cup$ Spec$\mathbb{C}[x, y]/(x-t) = $ Spec$\mathbb{C}[y]$. And since $\mathbb{V}(x(x-t)) = \{(0, z), (t, z) \mid z \in \mathbb{C}\}$, the maximal spectrum contains the ideals $(x, y-z), (x-t, y-z)$. But what else can I say? Is the scheme homeomorphic to the line as well, or something like it?
Because $V(x(x-t)) = V(x) \cup V(x-t)$, we have the equality
$$\text{Spec}\frac{\mathbb{C}[x,y]}{(x(x-t))} = \text{Spec}\frac{\mathbb{C}[x,y]}{(x)} \cup \text{Spec}\frac{\mathbb{C}[x,y]}{(x-t)} \\ = \text{Spec} \mathbb{C}[u] \cup \text{Spec} \mathbb{C}[v] \\ = \mathbb{A}^{1} \coprod \mathbb{A}^{1}.$$
This holds for any choice of $t \in \mathbb{C}$. But when $t=0$, Spec$\mathbb{C}[x,y]/(x^{2}) = \{(x)\}$, and so the point $(x)$ is dense.