## Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline {\mathcal {M}}_{8}$

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## Abstract:

For a curve $C$ of genus $6$ or $8$ and a torsion bundle $\eta$ of order $\ell$ we study the vanishing of the space of global sections of the twist $E_C \otimes \eta$ of the rank $2$ Mukai bundle $E_C$ of $C$. The bundle $E_C$ was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plücker embedding.

Globalizing the vanishing condition, we obtain divisors on the moduli spaces $\overline {\mathcal {R}}_{6,\ell }$ and $\overline {\mathcal {R}}_{8,\ell }$ of pairs $[C, \eta ]$. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that $\overline {\mathcal {R}}_{8,3}$ is of general type.

## References

- Dan Abramovich, Alessio Corti, and Angelo Vistoli,
*Twisted bundles and admissible covers*, Comm. Algebra**31**(2003), no. 8, 3547–3618. Special issue in honor of Steven L. Kleiman. MR**2007376**, DOI 10.1081/AGB-120022434 - Marian Aprodu and Jan Nagel,
*Koszul cohomology and algebraic geometry*, University Lecture Series, vol. 52, American Mathematical Society, Providence, RI, 2010. MR**2573635**, DOI 10.1090/ulect/052 - Ingrid Bauer and Fabrizio Catanese,
*The rationality of certain moduli spaces of curves of genus 3*, Cohomological and geometric approaches to rationality problems, Progr. Math., vol. 282, Birkhäuser Boston, Boston, MA, 2010, pp. 1–16. MR**2605163**, DOI 10.1007/978-0-8176-4934-0_{1} - Ingrid Bauer and Alessandro Verra,
*The rationality of the moduli space of genus-4 curves endowed with an order-3 subgroup of their Jacobian*, Michigan Math. J.**59**(2010), no. 3, 483–504. MR**2745749**, DOI 10.1307/mmj/1291213953 - Arnaud Beauville,
*Prym varieties and the Schottky problem*, Invent. Math.**41**(1977), no. 2, 149–196. MR**572974**, DOI 10.1007/BF01418373 - Arnaud Beauville,
*Vector bundles on curves and theta functions*, Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math., vol. 45, Math. Soc. Japan, Tokyo, 2006, pp. 145–156. MR**2310248**, DOI 10.2969/aspm/04510145 - Mira Bernstein,
*Moduli of curves with level structure*, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–Harvard University. MR**2699319** - Gregor Bruns,
*$\overline {\mathcal {R}}_{15}$ is of general type*, Algebra Number Theory**10**(2016), no. 9, 1949–1964. MR**3576116**, DOI 10.2140/ant.2016.10.1949 - Lucia Caporaso, Cinzia Casagrande, and Maurizio Cornalba,
*Moduli of roots of line bundles on curves*, Trans. Amer. Math. Soc.**359**(2007), no. 8, 3733–3768. MR**2302513**, DOI 10.1090/S0002-9947-07-04087-1 - Alessandro Chiodo,
*Towards an enumerative geometry of the moduli space of twisted curves and $r$th roots*, Compos. Math.**144**(2008), no. 6, 1461–1496. MR**2474317**, DOI 10.1112/S0010437X08003709 - Alessandro Chiodo, David Eisenbud, Gavril Farkas, and Frank-Olaf Schreyer,
*Syzygies of torsion bundles and the geometry of the level $\ell$ modular variety over $\overline {\scr {M}}_g$*, Invent. Math.**194**(2013), no. 1, 73–118. MR**3103256**, DOI 10.1007/s00222-012-0441-0 - Gavril Farkas,
*Koszul divisors on moduli spaces of curves*, Amer. J. Math.**131**(2009), no. 3, 819–867. MR**2530855**, DOI 10.1353/ajm.0.0053 - G. Farkas, S. Grushevsky, R. Salvati Manni, and A. Verra,
*Singularities of theta divisors and the geometry of $\scr {A}_5$*, J. Eur. Math. Soc. (JEMS)**16**(2014), no. 9, 1817–1848. MR**3273309**, DOI 10.4171/JEMS/476 - Gavril Farkas and Katharina Ludwig,
*The Kodaira dimension of the moduli space of Prym varieties*, J. Eur. Math. Soc. (JEMS)**12**(2010), no. 3, 755–795. MR**2639318**, DOI 10.4171/JEMS/214 - Gavril Farkas and Alessandro Verra,
*Moduli of theta-characteristics via Nikulin surfaces*, Math. Ann.**354**(2012), no. 2, 465–496. MR**2965251**, DOI 10.1007/s00208-011-0739-z - Mark L. Green,
*Koszul cohomology and the geometry of projective varieties*, J. Differential Geom.**19**(1984), no. 1, 125–171. MR**739785** - H. Lange, V. Mercat, and P. E. Newstead,
*On an example of Mukai*, Glasg. Math. J.**54**(2012), no. 2, 261–271. MR**2911367**, DOI 10.1017/S0017089511000577 - Shigeru Mukai,
*Curves and Grassmannians*, Algebraic geometry and related topics (Inchon, 1992) Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 19–40. MR**1285374** - D. Mumford,
*On the equations defining abelian varieties. I*, Invent. Math.**1**(1966), 287–354. MR**204427**, DOI 10.1007/BF01389737 - Michel Raynaud,
*Sections des fibrés vectoriels sur une courbe*, Bull. Soc. Math. France**110**(1982), no. 1, 103–125 (French, with English summary). MR**662131** - A. Verra, 2016. Private communication.

## Additional Information

**Gregor Bruns**- Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
- MR Author ID: 1190736
- Email: math@gregorbruns.eu
- Received by editor(s): November 1, 2016
- Received by editor(s) in revised form: March 14, 2017, and March 17, 2017
- Published electronically: June 7, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8359-8376 - MSC (2010): Primary 14H10, 14H45; Secondary 14E08, 14H40, 14K10
- DOI: https://doi.org/10.1090/tran/7239
- MathSciNet review: 3864379