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

Abstract

Let $X$ be a non-degenerate projective irreducible variety of dimension $n\ge 1$, degree $d$, and codimension $e\ge 2$ over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Let ${\beta }_{p,q}(X)$ be the $(p,q)$-th graded Betti number of $X$. M. Green proved the celebrating ${\mathcal{K}}_{p,1}$-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 ${\beta }_{e-1,1}(X)\ne 0$ 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 ${\mathbb{P}}^9$. In this paper, we completely classify varieties $X$ with nonvanishing ${\beta }_{e-1,1}(X)\ne 0$ 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 $\le n-1$.

18 pages