## A Geometric Reason to Care About Discrete Valuation Rings

Perhaps those that have taken an introductory commutative algebra class have come across the profoundly opaque definition of a discrete valuation ring (DVR), thinking to themselves "why have the math gods condemned me?" From a purely algebraic perspective, it is a remarkable that a ring $A$ is a valuation ring if and only if it is a domain so that for any element in the fraction field for $A$, $x$ or $x^{-1}$ is contained in $A$. Even further, if a valuation ring is Noetherian (and not a field) it is a DVR. Again, algebraically, there are a lot of reasons to care about these rings, but today isn't about algebra. While there are a huge number of equivalences we can speak of, but instead we will make use of the following definition: Definition:…