# Geometric meaning of Koszul modules

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" 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?

• You will have to do some digging yourself, but there is a paper of Green and Lazarsfeld in which they show how to construct non-zero Koszul cohomology classes from points in special position w.r.t a line bundle. 'not p+1 very ample' means that there are p+1 points in special position relative to L. The construction of the class is explicit and imho, geometric. Happy hunting.
– meh
Nov 17 '19 at 17:47
• Hi ! This is a result from Koh-Stillman's paper "Linear Syzygies and Line Bundles on an Algebraic Curve", see Prop. 3.6. You can also look at Chapter 4.4 of Aprodu-Nagel's book "Koszul Cohomology and Algebraic Geometry", in particular Theorem 4.36. Nov 24 '19 at 4:00
• @mkemeny But those propositions induce that $L$ does not satisfy $N_p$(in fact $N_{p-1}$ in my situation). This doesn't imply the nonvanishment of $K_{p,2}$, right? Dec 12 '19 at 23:57
• These two conditions are basically the same in our setting, see the top of page 8 of arxiv.org/pdf/1408.4164.pdf Dec 14 '19 at 0:35