This is related to a comment made in Mumford Algebraic Geometry I, Complex Projective Geometry.
"... in fact every $x\in X$ which is not fundamental and where $O_{x,X}$ is integrally closed in $\Bbb C(X)$ is actually a regular point."
$X$ above is a projective variety and $\Bbb C(X)$ denotes function field of $X$ over complex number. Assume $X,Y$ projective varieties s.t $f:X\to Y$ is a dominant rational map. The set of points of $x\in X$ s.t. fiber of $f(x)$ with codimension $>\dim(X)-\dim(Y)$ is called fundamental. It is clear that fundamental points are closed.
Regular local implies UFD which implies integrally closed.
$\textbf{Q:}$ How is not being fundamental and local ring integrally closed related to regularness?