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In M. Aprodu, G. Farkas - Koszul Cohomology and Applications to Moduli, arXiv:0811.3117 [math.AG], proof of Theorem(s) 4.5 (and 4.12), the authors constructed a semistable nodal curve $C^{\prime}$ of genus $2g-2d+3$ with $\delta=g-2d+3$ nodes starting from a smooth curve $C$ of genus $g$ and gonality $\displaystyle3\leq d\leq\left\lfloor\frac{g}{2}\right\rfloor+1$ (M. Aprodu - Remarks on syzygies of $d$-gonal curves, arXiv:math/0412134v3 [math.AG]) and identifying $\delta$ pairs of points on $C$ in general position.

They prove that the gonality of $C^{\prime}$ is $g-d+3$: until this point the proof is right for me!

After this, they apply Theorem 4.6 and state that $K_{g-d+1,1}\left(C^{\prime},\omega_{C^{\prime}}\right)=0$; but this last theorem require that $g=2d-1\geq7$: what do I am missing?

Possibile answer: does Theorem 4.6 holds for $g\geq5$? (cfr. M. Aprodu, Proposition 7.) If yes, "I win".

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    $\begingroup$ The Aprodu's condition works for $d\geq3$ (cfr. also M. Aprodu, J. Nagel - Koszul Cohomolgy and Algebraic Geometry), on the other hand, for $d=2$ one has an hyperelliptic curve, for which the Green's conjecture does not work; and the Aprodu - Farkas Theorem 4.6 works for $g\geq5$ (cfr. Aprodu Proposition 8 and Aprodu - Nagel Theorem 7.6). $\endgroup$
    – Armando j18eos
    Commented Feb 10, 2022 at 15:23

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