Rational fibrations of \(\overline M_{5,1}\) and \(\overline M_{6,1}\) (Q418917)

The author constructs rational maps from \(\overline{M}_{5,1}\) and \(\overline{M}_{6,1}\) to lower-dimen\-sional moduli spaces. Geometric divisors that generate extremal rays of the effective cones for these spaces are identified. In particular, it is proved that if \(D_6\) is the closure of the locus of pointed curves \((C, p)\in M_{6,1}\) possessing a \(g^2_6\) \(\mathcal L\) and a point \(p'\in C\) such that \(h^0(C, {\mathcal L}-p -p')\geq 2\), then \(D_6\) generates an extremal ray of \(\overline{NE}^1(\overline{M}_{6,1})\).