Are curves with maximal Clifford index Brill-Noether general? 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.
 A: The answer to (Q1) is definitely no. The first example is in genus 4: a (smooth) complete intersection of a quadric cone and a cubic surface in $\mathbb{P}^3$ has maximal Clifford index (= 1), but is not Brill-Noether general: it has a unique $g^1_3$ whose double is the canonical divisor.
The answer to (Q2) should be yes. I recommend the paper The Clifford dimension of a projective curve,  by Eisenbud, Lange, Martens, Schreyer (Compositio Math. 72 (1989), no. 2, 173–204). They conjecture that the Clifford index comes from a $g_r^d$ with $r>1$ only for plane curves and in some very particular cases, where the Clifford index is one less than the maximal one. This would imply a positive answer to (Q2). Of course (Q2) is much weaker, so there may very well be a proof independent of the conjecture -- but I do not see it at the moment.
A: 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.
