The moduli and the global period mapping of surfaces with : a counterexample to the global Torelli problem
The moduli space of totally marked degree two rational maps
A rational map ϕ: ℙ¹ → ℙ¹ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps can be parametrized by an affine open subset of (ℙ¹)⁵. We consider the natural action of SL₂ on induced from the action of SL₂ on (ℙ¹)⁵ and prove that the quotient space exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over ℤ[1/2].
The Singular Points on the Moduli Spaces for Smooth Curves.
Torsion in elliptic curves over