Some remarks on the κ_<i>p</i>,1 theorem

초록

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 beta(p,q)(X) be the (p, q)th graded Betti number of X. Green proved the celebrating kappa(p,1)-theorem about the vanishing of beta(p,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 beta(e-1,1)(X). It is clear that beta(e-1,1)(X) not equal 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 P-9. In this paper, we completely classify varieties X with nonvanishing beta(e-1,1)(X) not equal 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.

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