Birational contraction of genus two tails in the moduli space of genus four curves I

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We show that for α∈ (2/3, 7/10), the log canonical model M₄(α) of the pair (M₄, αδ) is isomorphic to the moduli space M₄hs of h-semistable curves, and that there is a birational morphism Ξ: M₄hs → M₄(2/3) that contracts the locus of curves C₁∪ₚ C₂ consisting of genus two curves meeting in a node p such that p is a Weierstrass point of C₁ or C₂. To obtain this morphism, we construct a compact moduli space M₂,₁hs of pointed genus two curves that have nodes, ordinary cusps and tacnodes as singularity, and prove that it is isomorphic to Rulla's flip constructed in his thesis.