Minimal Free resolution of the ideals What is the motivation to study the minimal free resolution of the ideals? Which geometrical information can we get from $res(I)$ for The variety of $I$?
 A: The most basic motivation for studying resolutions is computing the Hilbert function/polynomial of $Y=V(I)$. The geometric information in this polynomial is essentially the dimension, the embedded degree of $Y$ and the arithmetic genus of $Y$.
More refined information can be found by looking at the betti numbers of $I$. By definition, these are the ranks of the free modules appearing in the resolution and thus tell us about the syzygies of $I$, that is, the relations between the generators of $I$. A standard example (see Eisenbud's book) is the case of 7 points in $\mathbb{P}^3$: All such configurations of points have the same Hilbert polynomial, but the graded Betti numbers of the ideal of the points determine whether the points lie on a rational normal curve or not. 
For curves, there are also interesting relations between the betti numbers of a canonically embedded curve and the Clifford index of the curve (which is a geometrical invariant of the curve). This is the main content of Green's conjecture, which is still open. 
For more info on this, take a look at Eisenbud's excellent book 'The Geometry of Syzygies' or Wiegand's article 'What is.. a Syzygy?'
