By the Brill-Noether Theorem, a general curve $C$ of genus $g\geq2$ has maximal Clifford index $\lfloor \frac{g-1}{2}\rfloor$. Hence a very naive question is:


(Q1) Is a curve with maximal Clifford index a *general* curve (in the sense of Brill-Noether theory) ? And if not, what can be said about such a curve (when non-general) ?


Of course a similar setting holds by considering the gonality instead of the Clifford index, but I guess that in this situation the two are completely equivalent, that is:


(Q2) Is the gonality of a curve with *maximal* Clifford index $c$ always equal to $c+2$ ?


Edit: by curve I understand an irreducible curve.



Are curves with maximal Clifford index Brill-Noether general?
Q2) as stated is obviously true, because if the curve had a pencil of degree less than or equal to c+1, then the Clifford index would be at most c-1 (the clifford index of this pencil). So you probably meant to state the converse statement, which asks if a curve of maximal gonality has maximal Clifford index.


At least if the genus is odd, the answer to this question is yes. In this case, a curve has maximal gonality if and only if it has maximal Clifford index. This comes out as a corollary of a beautiful paper by Hirschowitz-Ramanan *"New evidence for Green's conjecture on syzygies of canonical curves"*. One knows, by the generic Green's conjecture (proved by Voisin), that a curve has nonmaximal Clifford index iff it has an extra syzygy, in the sense of Green. On the other hand, one can compare the divisor of curves in $\mathcal{M}\_g$ with extra syzygies with the *Hurwitz* divisor of curves with nonmaximal gonality. The two divisors can be shown to be the same using direct computation of their classes, and this gives the result. For more details, see the Hirschowitz-Ramanan paper above.


