Some remarks on the K₏, ₁ Theorem

Let X be a non-degenerate projective irreducible variety of dimension n 1, degree d, and codimension e 2 over an algebraically closed field K of characteristic 0. Let ₏, ₐ (X) be the (p, q) -th graded Betti number of X. M. Green proved the celebrating K₏, ₁-theorem about the vanishing of ₏, ₁ (X) for high values for p and potential examples of nonvanishing graded Betti numbers. Later, Nagel-Pitteloud and Brodmann-Schenzel classified varieties with nonvanishing ₄-₁, ₁ (X). It is clear that ₄-₁, ₁ (X) 0 when there is an (n+1) -dimensional variety of minimal degree containing X, however, this is not always the case as seen in the example of the triple Veronese surface in P⁹. In this paper, we completely classify varieties X with nonvanishing ₄-₁, ₁ (X) 0 such that X does not lie on an (n+1) -dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties whose Picard number is n-1.