Some remarks on the Kp,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 βp,q (X) be the $(p,q)$-th graded Betti number of X. M. Green proved the celebrating Kp,1-theorem about the vanishing of βp,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 βe-1,1(X). It is clear that βe-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 βe-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.