Does anyone know a good reference to understand the historical background of Castelnuovo-Mumford regularity? 


I know the backgound for the modern commutative-algebra approach (using free graded resolutions and Betti numbers) but I'd like to know the geometric motivations that led to the sheaf cohomology definition of $m$-regular sheaf. 


Mumford, in his *Lectures on curves on an algebraic surface* and in a later article ascribes the definition of regular sheaves to Castelnuovo. I'd like to deepen in this way.



Historical refererences for Castelnuovo-Mumford regularity
You might like looking at Eisenbud, Green and Harris "[Cayley-Bacharach theorems and conjectures](http://www.ams.org/journals/bull/1996-33-03/S0273-0979-96-00666-0/home.html)". They start with classical theorems of Euclidean geometry, such as Pappus and Pascal's theorem, and relate them to questions in commutative algebra regarding in what degree the saturated homogenous ideals of various sets in $\mathbb{P}^2$ are generated. This is to say, they relate them to describing the $0$-syzygies of the ideal. CM-regularity is a measure of the complexity of all syzygies.


If you more specifically want to know why people historically studied higher syzygies, I've heard good things about Eisenbud's "[Geometry of syzygies](http://www.msri.org/people/staff/de/ready.pdf)", although I haven't read it.


