I don't understand a claim of Lazarsfeld in his paper " A sampling of vector bundle techniques in the study of linear series. The setting is as follows: There is a very ample line bundle $L$ on a curve $C$, which embeds $C$ as a projectively normal curve.
In the proof of Proposition 2.4.2, Lazarsfeld claims that $C$ is NOT (set theoretically) cut out by quadrics if and only if there exists a non-split extension
$0\to K_C\otimes L^{-1}\to E\to L\to 0$ such that
$i)$ The extension does not correspond to $H^0(L^2(-p))^\vee$ for some $p\in C$. (The vector space $Ext^1(L,K_C\otimes L^{-1})$ is isomorphic to $H^0(L^2)^\vee$ and $H^0(L^2(-p))^\vee$ being an element of this vector space corresponds to an extension of $K_C\otimes L^{-1}$ by $L$.)
$ii)$ The connecting morphism $H^0(L)\to H^1(K_C\otimes L^{-1})$ has rank one.
I don't understand why this statement holds. Can anybody help me with that?