  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 $\beta_{e-1,1}(X) \neq 0$
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 $\beta_{e-1,1}(X) \neq 0$ 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 $\le n-1$.
