Some remarks on the Kp,1$\mathcal {K}_{p,1}$ theorem

Abstract

AbstractLet be a non‐degenerate projective irreducible variety of dimension , degree , and codimension over an algebraically closed field of characteristic 0. Let be the th graded Betti number of . Green proved the celebrating ‐theorem about the vanishing of for high values for and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing . It is clear that when there is an ‐dimensional variety of minimal degree containing , however, this is not always the case as seen in the example of the triple Veronese surface in .In this paper, we completely classify varieties with nonvanishing such that does not lie on an ‐dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties, whose Picard number is .