Ordinary Double Point Characterization I'm reading Qing Liu's Algebraic Geometry and Arithmetic Curves in an attempt to understand Artin-Winters' proof of Deligne-Mumford's semi-stable reduction theorem. I'm having a bit of trouble understanding the algebraic characterization of ordinary double points. In particular, letting $X$ be a reduced curve over an algebraically closed field $k$, and letting $\pi:X'\to X$ be the normalization morphism, we call a closed point $x\in X$ an ordinary double point if $\pi^{-1}(x)$ consists of 2 points, and we have$$\dim_k\left(\mathcal{O}'_{X,x}/\mathcal{O}_{X,x}\right) = [k(x):k] = 1$$where $\mathcal{O}'_{X,x}$ is the integral closure of $\mathcal{O}_{X,x}$ in $\operatorname{Frac}(\mathcal{O}_{X,x})$. I'm having trouble understanding what the dimension of the quotient $\mathcal{O}'_{X,x}/\mathcal{O}_{X,x}$ as a $k$-vector space means (geometrically, say) and, in particular, what it means for it to be 1. Any help is appreciated! Thank you!
 A: The dimension of $\mathcal{O}_{X,x}'/\mathcal{O}_{X,x}$ is "how many" extra functions you need to be able to tell the germs of $X'$ through the two points $x_1,x_2$ on the normalization living above $x$ apart. As the normalization map is given by some undue gluing of subvarieties and tangent spaces, it measures how much stuff got glued together. In the case of an ordinary double point, the answer is that two points got glued together but none of their tangent directions - the function that gets added to form the integral closure is one which distinguishes the points $x_1$ and $x_2$.
It may also be helpful to hear a little bit about other better definition of a node (aka ordinary double point). Vakil and the Stacks Project define a node (at least in the algebraically closed case) to be a singular point on a curve which has $\mathcal{O}_{X,x}^\wedge\cong k[[x,y]]/(xy)$ as topological rings. In my opinion, this is much clearer to understand. It turns out that your definition in terms of the number of preimages and $\delta=\dim \mathcal{O}_{X,x}'/\mathcal{O}_{X,x}$ is equivalent to this - the main ingredient is Stacks 0C3W saying that it doesn't matter where we compute $\delta$ - the local ring, the strict henselization, or the completion will all give the same result.
