TITLE: Why considering schemes over discrete valuation rings?
QUESTION [7 upvotes]: For many times, I find people working on schemes over DVRs, and prove theorems on such setting. For example, my latest experience is the "semi-stable reduction theorem" by Kempf, Knudsen, Mumford and Saint-Donat: they proved semi-stable reduction theorem first over $\mathbb{C}$, then turned to the situation over DVR. 
It is "natural" for me to work over $\mathbb{C}$, or over character $p$, or even over arbitrary schemes. But why people like to choose DVRs in particular? This question would break to two parts: 
(1) Is there any technical advantage to work over DVR?
(2) What is the importance to consider this special case? Does this come from the interests of number theory? I have no clue of that.

REPLY [5 votes]: (1) What are the technical advantages of working over a DVR?
They are many. DVR are very simple rings, they are both local and principal.
The principality in particular means that the flatness (which is a fundament and not sue say to grasp property in algebraic geometry) of a module or algebra over that sort of ring becomes something much simpler to understand, to check and to use, namely the propriety of being torsion-free. Also, a DVR is integrally closed, and that's a property that is very useful for certain type of reasoning in algebraic geometry.
(2) The interest of working with a DVR in algebraic geometry does not come primarily from number theory. For example, you may have seen the valuative criterion of properness: to check if a morphism $f: X \rightarrow Y$ is proper, 
under certain mild hypothesis, you only have to check that the pull-back of this morphism over any spectrum of DVR $Y'$ is. This reduces the verification of an important relative property over an arbitrary base to the case of much simpler base, see (1), and is a crucial tool in algebraic geometry.