## Algebra & Number Theory

### $\overline{\mathscr{R}}_{15}$ is of general type

Gregor Bruns

#### Abstract

We prove that the moduli space $ℛ ¯15$ of Prym curves of genus $15$ is of general type. To this end we exhibit a virtual divisor $D¯15$ on $ℛ¯15$ as the degeneracy locus of a globalized multiplication map of sections of line bundles. We then proceed to show that this locus is indeed of codimension one and calculate its class. Using this class, we can conclude that $Kℛ ¯ 15$ is big. This complements a 2010 result of Farkas and Ludwig: now the spaces $ℛ¯g$ are known to be of general type for $g ≥ 14$.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 9 (2016), 1949-1964.

Dates
Revised: 19 April 2016
Accepted: 30 August 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842610

Digital Object Identifier
doi:10.2140/ant.2016.10.1949

Mathematical Reviews number (MathSciNet)
MR3576116

Zentralblatt MATH identifier
1351.14018

#### Citation

Bruns, Gregor. $\overline{\mathscr{R}}_{15}$ is of general type. Algebra Number Theory 10 (2016), no. 9, 1949--1964. doi:10.2140/ant.2016.10.1949. https://projecteuclid.org/euclid.ant/1510842610

#### References

• E. Ballico, “A remark on linear series on general $k$-gonal curves”, Boll. Un. Mat. Ital. A $(7)$ 3:2 (1989), 195–197.
• E. Ballico, C. Casagrande, and C. Fontanari, “Moduli of Prym curves”, Doc. Math. 9 (2004), 265–281.
• A. Beauville, “Prym varieties and the Schottky problem”, Invent. Math. 41:2 (1977), 149–196.
• M. Bernstein, Moduli of curves with level structure, Ph.D. thesis, Harvard University, 1999, hook http://search.proquest.com/docview/304503454 \posturlhook.
• R. Biggers and M. Fried, “Irreducibility of moduli spaces of cyclic unramified covers of genus $g$ curves”, Trans. Amer. Math. Soc. 295:1 (1986), 59–70.
• F. Catanese, “Homological algebra and algebraic surfaces”, pp. 3–56 in Algebraic geometry (Santa Cruz, 1995), edited by J. Kollár et al., Proc. Sympos. Pure Math. 62, American Mathematical Society, Providence, RI, 1997.
• A. Chiodo, D. Eisenbud, G. Farkas, and F.-O. Schreyer, “Syzygies of torsion bundles and the geometry of the level $\ell$ modular variety over $\overline{\mathscr{M}}\sb g$”, Invent. Math. 194:1 (2013), 73–118.
• D. Eisenbud and J. Harris, “Irreducibility and monodromy of some families of linear series”, Ann. Sci. École Norm. Sup. $(4)$ 20:1 (1987), 65–87.
• D. Eisenbud and J. Harris, “Irreducibility of some families of linear series with Brill–Noether number $-1$”, Ann. Sci. École Norm. Sup. $(4)$ 22:1 (1989), 33–53.
• G. Farkas, “Koszul divisors on moduli spaces of curves”, Amer. J. Math. 131:3 (2009), 819–867.
• G. Farkas and K. Ludwig, “The Kodaira dimension of the moduli space of Prym varieties”, J. Eur. Math. Soc. 12:3 (2010), 755–795.
• G. Farkas and A. Verra, “Prym varieties and moduli of polarized Nikulin surfaces”, Adv. Math. 290 (2016), 314–328.
• G. Farkas, S. Grushevsky, R. Salvati Manni, and A. Verra, “Singularities of theta divisors and the geometry of ${A}\sb 5$”, J. Eur. Math. Soc. 16:9 (2014), 1817–1848.
• J. Harris, “On the Severi problem”, Invent. Math. 84:3 (1986), 445–461.
• A. Iliev and D. Markushevich, “Quartic 3-fold: Pfaffians, vector bundles, and half-canonical curves”, Michigan Math. J. 47:2 (2000), 385–394.
• D. Mumford, “On the equations defining abelian varieties. I”, Invent. Math. 1 (1966), 287–354.