We study some particular loci inside the moduli space 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′)≥2). We show that the bielliptic locus is not a totally geodesic subvariety of A_g if g≥4 (while it is for g=3, see [16]) and that the bihyperelliptic locus is not totally geodesic in A_g if g≥3g′. We also give a lower bound for the rank of the second gaussian map on the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.
On the bielliptic and bihyperelliptic loci
Frediani, P.
;Porru, P.
2020-01-01
Abstract
We study some particular loci inside the moduli space 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′)≥2). We show that the bielliptic locus is not a totally geodesic subvariety of A_g if g≥4 (while it is for g=3, see [16]) and that the bihyperelliptic locus is not totally geodesic in A_g if g≥3g′. We also give a lower bound for the rank of the second gaussian map on the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
