Syzygies of Veronese embeddings (Q2709816)

The authors prove that if the Veronese embedding \({\mathbb P}^n\) to \({\mathbb P}^{n+d \choose n}\) satisfies Green's property \(N_p\), then either \(n = 2\), \(d=2\), \(p \geq 0\) or \(n \geq 3\), \(d = 2\), \(p \leq 5\) or \(n \geq 2\), \(d \geq 3\), \(p \leq 3d-3\). They raise the conjecture that the converse is also true. That is the case if \(d = 2\) [\textit{T. Jozefiak, P. Pragacz} and \textit{J. Weyman}, Asterisque 87-88, 109-189 (1981; Zbl 0488.14012)] or \(n = 2\) or \(n = d = 3\) [\textit{M. L. Green}, Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)].