Terminology: $\operatorname{Spec}A\subset \mathbb{A}^2$

Question

I came accross the following definition of an affine plane conic: it is $\operatorname{Spec}A\subset \mathbb{A}^2$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors.

Now, $\operatorname{Spec}A$ is the set of all prime ideals in $A$ which is the same as the set of all prime ideals in $k[x,y]$ containing the ideal $(f)$. Please help me to understand why/how this set is a subset of $\mathbb{A}^2$. Is it a jargon? What I mean is that of course prime ideals in $k[x,y]$ correspond bijectively to affine varieties in $\mathbb{A}^2$ by the Nullstellensatz, but by themselves they don't lie in $\mathbb{A}^2$, do they?

Answer 1

$\DeclareMathOperator{\Spec}{Spec}$ If you wanted to be a real stickler, I think you would say that there is a closed embedding $\varphi: \Spec(A) \to \mathbb{A}^2$. This follows by contravariance: we have a surjective map $\pi: k[x,y] \to k[x,y]/(f) = A$ which induces the map $$ \varphi: \Spec(A) \to \Spec(k[x,y]) = \mathbb{A}^2 \, . $$ Most of the time you would identify $\Spec(A)$ with its image under $\varphi$ and consider it a closed subscheme of $\mathbb{A}^2$.

Answer 2

There are two definitions of $\Bbb A^n$:

$$\Bbb A_k^n=\{(a_1,\dots,a_n):a_i\in k\}$$

and

$$\Bbb A_k^n=\operatorname{Spec}k[x_1,\dots,x_n].$$

The former is what you'd usually encounter in a first course in algebraic curves, and the latter is what you use once you start doing scheme theory. Using the latter, the definition given makes perfect sense.