Hartshorne remarks that is is something of a mystery as to why the algebraic condition of flatness on the structure sheaves gives a good definition of a family (see below). Are there any known enlightening explanations that help serve to unravel this mystery? Below is Hartshorne's introductory motivation to flat families containing said remark:

For many reasons it is important to have a good notion of an algebraic family of varieties or schemes. The most naive definition would be just to take the fibres of a morphism. To get a good notion, however, we should require that certain numerical invariants remain constant in a family, such as the dimension of the fibres. It turns out that if we are dealing with non- singular (or even normal) varieties over a field, then the naive definition is already a good one. Evidence for this is the theorem (9.13) that in such a family, the arithmetic genus is constant.

On the other hand, if we deal with nonnormal varieties, or more general schemes, the naive definition will not do. So we consider a flat family of schemes, which means the fibres of a flat morphism, and this is a very good notion. Why the algebraic condition of flatness on the structure sheaves should give a good definition of a family is something of a mystery. But at least we will justify this choice by showing that flat families have many good properties, and by giving necessary and sufficient conditions for flatness in some special cases. In particular, we will show that a family of closed subschemes of projective space (over an integral scheme) is flat if and only if the Hilbert polynomials of the fibres are the same. -- Hartshorne, Algebraic Geometry, 1977, III.9.5, p. 256

flatover $S$, then for any $f:S' \rightarrow S$ the exact sequence $0 \rightarrow I \rightarrow O_X \rightarrow j_{\ast}O_Z \rightarrow 0$ has $S$-flat right term, so $f^{\ast}I \rightarrow O_{X'}$ isinjective. Thus, ideal of $Z' \hookrightarrow X'$ is the "abstract" pullback $f^{\ast}I$ treating $I$ as "abstract" $O_X$-module. This is important. $\endgroup$ – BCnrd Aug 10 '10 at 16:11equivalentto flatness of $f$ at $x$. A great non-trivial example is q-finite map between smooth varieties of same dimension. Serre told me that the geometric significance of flatness was entirely due to Grothendieck (Serre invented it as a purely algebraic device for GAGA). $\endgroup$ – BCnrd Aug 10 '10 at 16:18