Some remarks on the K_p,1 theorem

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