  We study the minimal free resolution of the Veronese modules of the
polynomial ring in n variables, by giving a formula for the Betti numbers in
terms of the reduced homology of some skeleton of a simplicial complex. We
characterize when they are Cohen-Macaulay and we give a sufficient condition
for the linearity of their minimal free resolution. We also conjecture that in
2 variables the Veronese modules have always pure resolutions. In addition, we
give a closed formula for their Hilbert series. As an application of our
results, we calculate the complete Betti diagrams of the Veronese subrings in
three variables with degree 4 and 5, and in four variables with degree 3.
