×

Syzygies of Prym and paracanonical curves of genus 8. (Syzygies de Prym et courbes paracanoniques de genre 8.) (English. French summary) Zbl 1407.13012

A general canonical curve is a general curve embedded in the projective space via the sections of the canonical bundle. A general paracanonical curve is a general curve embedded in the projective space via the sections of the canonical bundle twisted by an \(\ell\)-torsion line bundle. The classical Green’s Conjecture [C. Voisin, J. Eur. Math. Soc. (JEMS) 4, No. 4, 363–404 (2002; Zbl 1080.14525); Compos. Math. 141, No. 5, 1163–1190 (2005; Zbl 1083.14038)] predicts that the syzygies of a general canonical curve are natural. Similarly the Prym-Green Conjecture [G. Farkas and K. Ludwig, J. Eur. Math. Soc. (JEMS) 12, No. 3, 755–795 (2010; Zbl 1193.14043); A. Chiodo et al., Invent. Math. 194, No. 1, 73–118 (2013; Zbl 1284.14006)] predicts that the resolution of a general paracanonical curve is natural.
The Prym-Green Conjecture has been known to hold in various ranges of \(g\) and \(\ell\) [E. Colombo and P. Frediani, Bull. Lond. Math. Soc. 45, No. 5, 1031–1040 (2013; Zbl 1327.14130); G. Farkas and M. Kemeny, Invent. Math. 203, No. 1, 265–301 (2016; Zbl 1335.14009); Duke Math. J. 166, No. 6, 1103–1124 (2017; Zbl 1368.14053)]. In this paper, the authors show that the Prym-Green Conjecture surprisingly fails in genus eight. They indeed provide several geometric explanations for this failure, which may shed some light on possible other exceptions to the conjecture for genera with high divisability by two.

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
14H10 Families, moduli of curves (algebraic)
PDFBibTeX XMLCite
Full Text: arXiv Link