Syzygies of the Veronese modules
  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. 
Ornella Greco and Ivan Martino
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