## Duke Mathematical Journal

### The Prym–Green conjecture for torsion line bundles of high order

#### Abstract

Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove that the minimal resolution of a general paracanonical curve $C$ of odd genus $g$ and order $\ell\geq\sqrt{\frac{g+2}{2}}$ is natural, thus proving the Prym–Green conjecture. In the process, we confirm the expectation of Barth and Verra concerning the number of curves with $\ell$-torsion line bundle in a linear system on a special K3 surface.

#### Article information

Source
Duke Math. J. Volume 166, Number 6 (2017), 1103-1124.

Dates
Revised: 15 July 2016
First available in Project Euclid: 16 December 2016

https://projecteuclid.org/euclid.dmj/1481879046

Digital Object Identifier
doi:10.1215/00127094-3792814

Subjects
Primary: 14H
Secondary: 14H10: Families, moduli (algebraic)

#### Citation

Farkas, Gavril; Kemeny, Michael. The Prym–Green conjecture for torsion line bundles of high order. Duke Math. J. 166 (2017), no. 6, 1103--1124. doi:10.1215/00127094-3792814. https://projecteuclid.org/euclid.dmj/1481879046.

#### References

• [1] M. Aprodu and G. Farkas, The Green conjecture for smooth curves lying on arbitrary $K3$ surfaces, Compos. Math. 147 (2011), 839–851.
• [2] W. Barth and A. Verra, “Torsion on $K3$-sections” in Problems in the Theory of Surfaces and Their Classification (Cortona, 1988), Sympos. Math. 32, Academic Press, Boston, 1991, 1–24.
• [3] A. Chiodo, D. Eisenbud, G. Farkas, and F.-O. Schreyer, Syzygies of torsion bundles and the geometry of the level $\ell$ modular variety over $\overline{\mathcal{M}}_{g}$, Invent. Math. 194 (2013), 73–118.
• [4] A. Chiodo and G. Farkas, Singularities of the moduli space of level curves, to appear in J. Eur. Math. Soc., preprint, arXiv:1205.0201.
• [5] I. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), 2599–2630.
• [6] G. Farkas and M. Kemeny, The generic Green-Lazarsfeld Secant conjecture, Invent. Math. 203 (2016), 265–301.
• [7] G. Farkas and K. Ludwig, The Kodaira dimension of the moduli space of Prym varieties, J. Eur. Math. Soc. 12 (2010), 755–795.
• [8] G. Farkas and A. Verra, Moduli of theta-characteristics via Nikulin surfaces, Math. Ann. 354 (2012), 465–496.
• [9] M. Green, Koszul cohomology and the cohomology of projective varieties, J. Differential Geom. 19 (1984), 125–171.
• [10] G. Hardy and E. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Sci., New York, 1979.
• [11] D. Huybrechts and M. Kemeny, Stable maps and Chow groups, Doc. Math. 18 (2013), 507–517.
• [12] D. Huybrechts and M. Lehn, The Geometry of the Moduli Space of Sheaves, Cambridge Univ. Press, Cambridge, 2010.
• [13] R. Lazarsfeld, Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), 299–307.
• [14] J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Ergeb. Math. Grenzgeb. 73, Springer, New York, 1973.
• [15] D. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), 105–121.
• [16] M. Schütt and T. Shioda, “Elliptic surfaces” in Algebraic Geometry in East Asia (Seoul, 2008), Adv. Stud. Pure Math. 60, Math. Soc. Japan, Tokyo, 2010, 51–160.
• [17] T. Shioda, “Correspondence of elliptic curves and Mordell-Weil lattices of certain elliptic $K3$ surfaces” in Algebraic Cycles and Motives, Cambridge Univ. Press, Cambridge, 2007, 319–339.
• [18] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2016.
• [19] B. van Geemen and A. Sarti, Nikulin involutions on $K3$ surfaces, Math. Zeitschrift 255 (2007), 731–753.
• [20] C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a $K3$ surface, J. Eur. Math. Soc. 4 (2002), 363–404.
• [21] C. Voisin, Compos. Math. 141 (2005), 1163–1190.