$\DeclareMathOperator\Spec{Spec}$Suppose $X = \Spec A$ is a smooth affine variety over $\mathbb C$ and suppose $L/K$ is a finite extension of its function field. Let $Y = \Spec B$, where $B$ is the set of elements of $L$ which are integral over $A$. Is there a simple geometric/explicit description of $Y\to X$ if we know the field extension explicitly? This is a somewhat vague question so let me given an example of a kind of answer that I would consider geometric/explicit.

Suppose we know that $L$ is obtained by adjoining to $K$ a root of some irreducible polynomial $f\in K[t]$, i.e., we have $L \simeq K[t]/(f)$ as field extensions of $K$. Then, can we geometrically describe $Y\to X$ in terms of this $f$? If this is something well-known, a reference would be very much appreciated.

`\operatorname`

adds operator spacing automatically, so you can use $\operatorname{Spec} A$`\operatorname{Spec} A`

instead of $\text{Spec }A$`\text{Spec }A`

. If you will use an operator repeatedly, then you can define it once and for all as`\DeclareMathOperator\Spec{Spec}`

. I have edited accordingly. $\endgroup$8more comments