(\[mercat1\]) for bounded genus:

\[mercatcinci\] Mercat’s Conjecture (\[mercat1\]) holds for a general curve of genus $g\leq 16$.

The most beautiful case in the proof of Theorem \[mercatcinci\] is when $[C]\in \cM_{15}$ and $h^0(C, E)=5$. In order to show that $\mathcal{BN}_C(19, 5)=\emptyset$, one must argue that if $$\phi_L:C\stackrel{|L|}\longrightarrow \PP^6$$ is one of the embeddings of $C$ by a linear series $L\in W^6_{19}(C)$ residual to a pencil of minimal degree, then $\phi_L(C)$ cannot lie on $5$ independent quadric hypersurfaces in $\PP^6$. Note that $4=\mbox{dim } \mbox{Sym}^2 H^0(C, L)-h^0(C, L^{\otimes 2})$ independent quadrics containing $\phi_L(C)$ come automatically, and we show that the existence of a *fifth* quadric is a non-trivial condition in the moduli space $\cM_{15}$.

To recapitulate, the original prediction (\[mercat2\]) is not true when formulated in terms of the original Clifford index, but both (\[mercat1\]) and (\[mercat2\]) are still expected to hold for general curves in moduli! It is customary to view the Koszul geometry of a curve as *second order Brill-Noether theory*, in the sense that once all types of linear series $\mathfrak g^r_d$ on a curve have been prescribed, syzygies provide a finer analysis, distinguishing among curves with the same Brill-Noether behaviour. Our analysis lends some credence to the 