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

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 $K$ of characteristic 0. Let ${\beta }_{p,q}\left(X\right)$ be the $(p,q)$th graded Betti number of $X$. Green proved the celebrating $K_{p,1}$-theorem about the vanishing of ${\beta }_{p,1}\left(X\right)$ for high values for $p$ and potential examples of nonvanishing graded Betti numbers. Later, Nagel–Pitteloud and Brodmann–Schenzel classified varieties with nonvanishing ${\beta }_{e-1,1}\left(X\right)$. It is clear that ${\beta }_{e-1,1}\left(X\right)\ne 0$ 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 $P^9$.

In this paper, we completely classify varieties $X$ with nonvanishing ${\beta }_{e-1,1}\left(X\right)\ne 0$ 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 $\le n-1$.