Writing about spectrum of a ring Shafarevich in his Basic Algebraic geometry defines point $x \in \operatorname {Spec} A$ (which is some prime ideal of $A$) to be regular or simple if $\mathcal {O}_x$ (localization of $A$ by prime ideal $x$) is Noetherian and is a regular local ring.
Then he turns to the example of affine varieties and their coordinate ring and writes following about which I have a question:
Suppose that $A = k[X]$ and a point of $\operatorname {Spec} A$ corresponds to a prime ideal that is not maximal, that is, to an irreducible subvariety $Y \subset X$ of positive dimension. What is the geometric meaning of regularity of such a point? As the reader can easily check (using Theorem 2.13 of Section 3.2, Chapter 2), in this case, regularity means that $Y$ is not contained in the subvariety of singular points of $X$.
So how to prove that in this case $Y$ is not contained in the subvariety of singular points of $X$ ? Theorem that is mentioned here states:
Let $X$ be an affine variety, $x \in X$ a nonsingular point, and suppose that $u_1, . . . , u_n$ are regular functions on $X$ that form a system of local parameters at $x$. Then for $m ≤ n$, the subvariety $Y$ defined by $u_1 = · · · = u_m = 0$ is nonsingular at $x$, we have a $Y = (u_1, . . . , u_m)$ in some affine neighbourhood of $x$, and $u_{m+1}, . . . , u_n$ form a system of local parameters on $Y$ at $x$.
Unfortunately I can't see how it helps with our initial question. It states that if some point is regular on variety it is also regular on some subvariety containing it. While question asks on the opposite that if subvariety have some certain properties it has to contain some point that are regular in main variety containing it. So what I'm missing here?
I'm only in the process of studying all that stuff and my understanding of regular and singular points, tangent spaces and dimensions isn't very deep and most of it learnt not from book by Shafarevich. So any hints would be appreciated.