Assume $C$ is a smooth projective curve and $L$ is a line bundle on it. We say $L$ is $p$-very ample if for any effective divisor $D$ of degree $p+1$, the evaluation map $H^0(C,L)\to H^0(C,L\otimes\mathcal{O}_D)$ is surjective. Kemeny wrote in his paper "[The extremal secant conjecture for curves of arbitrary gonality][1]" that it is "rather straightforward" that if $L$ is not $p+1$-very ample then the Koszul cohomology $K_{p,2}(C;L)\not=0$. But I cannot see this, at least from the algebraic definition of the Koszul cohomology. 

Is there any geometric explanation for Koszul cohomology, in which $p+1$-very ampleness might be involved?


  [1]: https://www.cambridge.org/core/journals/compositio-mathematica/article/extremal-secant-conjecture-for-curves-of-arbitrary-gonality/7D607CAB936DCA8CFD54614075D9BB80