## Michigan Mathematical Journal

### On the Bielliptic and Bihyperelliptic Loci

#### Abstract

We study some particular loci inside the moduli space $\mathcal{M}_{g}$, namely the bielliptic locus (i.e. the locus of curves admitting a $2:1$ cover over an elliptic curve $E$) and the bihyperelliptic locus (i.e. the locus of curves admitting a $2:1$ cover over a hyperelliptic curve $C'$, $g(C')\geq 2$). We show that the bielliptic locus is not a totally geodesic subvariety of $\mathcal{A}_{g}$ if $g\geq 4$ (whereas it is for $g=3$, see [18]) and that the bihyperelliptic locus is not totally geodesic in $\mathcal{A}_{g}$ if $g\geq 3g'$. We also give a lower bound for the rank of the second Gaussian map at the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.

#### Article information

Source
Michigan Math. J., Advance publication (2020), 30 pages.

Dates
Revised: 17 August 2019
First available in Project Euclid: 31 January 2020

https://projecteuclid.org/euclid.mmj/1580439627

Digital Object Identifier
doi:10.1307/mmj/1580439627

#### Citation

Frediani, Paola; Porru, Paola. On the Bielliptic and Bihyperelliptic Loci. Michigan Math. J., advance publication, 31 January 2020. doi:10.1307/mmj/1580439627. https://projecteuclid.org/euclid.mmj/1580439627

#### References

• [1] R. D. Accola, Topics in the theory of Riemann surfaces, Springer, 2006.
• [2] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves I, Grundlehren Math. Wiss., 267, Wissenschaften, 1985.
• [3] M. Artebani, A. Laface, and G. P. Pirola, About the semiample cone of the symmetric product of a curve, Int. Math. Res. Not. 2017 (2016), no. 18, 5554–5576.
• [4] E. Ballico, G. Casnati, and C. Fontanari, On the geometry of bihyperelliptic curves, J. Korean Math. Soc. 44 (2007), no. 6, 1339–1350.
• [5] F. Bardelli and A. Del Centina, Bielliptic curves of genus three: canonical models and moduli space, Indag. Math. (N.S.) 10 (1999), no. 2, 183–190.
• [6] A. Broughton, The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology Appl. 37 (1990), no. 2, 101–113.
• [7] A. Calabri, C. Ciliberto, and R. Miranda, The rank of the second Gaussian map for general curves, Michigan Math. J. 60 (2011), no. 3, 545–559.
• [8] F. Catanese, M. Lönne, and F. Perroni, Irreducibility of the space of dihedral covers of the projective line of a given numerical type, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (2011), no. 3, 291–309.
• [9] F. Catanese, M. Lönne, and F. Perroni, Genus stabilization for the components of moduli spaces of curves with symmetries, J. Algebraic Geom. 3 (2016), no. 1, 23–49.
• [10] C. Ciliberto and R. Miranda, Gaussian maps for certain families of canonical curves, Complex projective geometry, London Math. Soc. Lecture Note Ser., 179, pp. 106–127, 1992.
• [11] E. Colombo and P. Frediani, Some results on the second Gaussian map for curves, Michigan Math. J. 58 (2009), no. 3, 745–758.
• [12] E. Colombo and P. Frediani, Siegel metric and curvature of the moduli space of curves, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1231–1246.
• [13] E. Colombo, P. Frediani, and A. Ghigi, On totally geodesic submanifolds in the Jacobian locus, Internat. J. Math. 26 (2015), no. 01, 1550005.
• [14] E. Colombo, P. Frediani, and G. Pareschi, Hyperplane sections of Abelian surfaces, J. Algebraic Geom. 21 (2012), no. 1, 183–200.
• [15] E. Colombo, G. P. Pirola, and A. Tortora, Hodge-Gaussian maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 30 (2001), no. 1, 125–146.
• [16] J. De Jong and R. Noot, Jacobians with complex multiplication. Arithmetic algebraic geometry (Texel, 1989), Prog. Math., 89, pp. 177–192, 1991.
• [17] P. Frediani, A. Ghigi, and M. Penegini, Shimura varieties in the Torelli locus via Galois coverings, Int. Math. Res. Not. 2015 (2015), no. 20, 10595–10623.
• [18] P. Frediani, M. Penegini, and P. Porru, Shimura varieties in the Torelli locus via Galois coverings of elliptic curves, Geom. Dedicata 181 (2016), no. 1, 177–192.
• [19] G. Gonzalez-Diez and W. J. Harvey, Moduli of Riemann surfaces with symmetry, London Math. Soc. Lecture Note Ser., 173, pp. 75–93, 1992.
• [20] M. Green, Infinitesimal methods in Hodge theory, Algebraic cycles and Hodge theory, pp. 1–92, Springer, 1994.
• [21] S. Grushevsky and M. Möller, Explicit formulas for infinitely many Shimura curves in genus 4, Asian J. Math. 22 (2018), no. 2, 381–390.
• [22] R. Miranda, Algebraic curves and Riemann surfaces, 5, American Mathematical Soc., 1995.
• [23] A. Mohajer and K. Zuo, On Shimura subvarieties generated by families of Abelian covers of $\mathbb{P}^{1}$, J. Pure Appl. Algebra 222 (2017), no. 4, 931–949.
• [24] B. Moonen, Linearity properties of Shimura varieties, I, J. Algebraic Geom. 7 (1998), no. 3, 539–568.
• [25] B. Moonen, Special subvarieties arising from families of cyclic covers of the projective line, Doc. Math. 15 (2010), 793–819.
• [26] B. Moonen and F. Oort, The Torelli locus and special subvarieties, Handbook of moduli: volume II, pp. 549–894, International Press, 2013.
• [27] G. D. Mostow, On discontinuous action of monodromy groups on the complex $n$-ball, J. Amer. Math. Soc. 1 (1988), no. 3, 555–586.
• [28] D. Mumford, A note of Shimura’s paper “discontinuous groups and Abelian varieties”, Math. Ann. 181 (1969), no. 4, 345–351.
• [29] F. Oort, Canonical liftings and dense sets of CM-points, Arithmetic geometry (Cortona, 1994) 37 (1997), 228–234.
• [30] F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves, Journées de Géometrie Algébrique d’Angers 1979 (1979), 157–204.
• [31] M. Penegini, Surfaces isogenous to a product of curves, braid groups and mapping class groups, Beauville surfaces and groups, pp. 129–148, Springer, 2015.
• [32] J. C. Rohde, An introduction to Hodge structures and Shimura varieties, Cyclic coverings, Calabi–Yau manifolds and complex multiplication, pp. 11–57, Springer, 2009.
• [33] G. Shimura, On purely transcendental fields automorphic functions of several variable, Osaka J. Math. 1 (1964), no. 1, 1–14.
• [34] J. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 (1987), 843–871.