Ornella Greco: Syzygies of the Veronese Modules

After defining the Veronese subrings, $S^{(d)}$, of a polynomial ring, $S=K[x_1,\dots,x_n]$,  I am going to give a summary of  some results on minimal free resolutions of Veronese rings. Then I will concentrate on the  minimal free resolution of the Veronese modules, defined as $S_{n,d,k}=\oplus_{i\geq 0}S_{k+id}$, and I will give  a formula for their Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. As applications, I will characterize the Cohen-Macaulayness of $S_{n,d,k}$ and the linearity of the resolution. I will give a formula for the Hilbert series of such modules and, as a consequence, this will lead to a complete description of the Betti tables of the Veronese modules that have pure resolutions.