Bergman kernel and period map for curves

Notes on hyperelliptic mapping class groups

“…Proof. (i): This is well known (see, for instance, item (h) of Theorem 1.4 in [18]) but also an almost immediate consequence of item (i) of Proposition 4.1.…”

“…the short exact sequence (15)) with the natural epimorphism Pϒ(S γ , P) → Pϒ(S γ , P ∪ {Q}) (cf. the last one of the short exact sequences (18)). There is a Birman short exact sequence:…”

“…The surface S γ is obtained from S γ by filling in the two punctures bounded by γ , so that S γ ∼ = S g−1 . By the bottom short exact sequence (18), there is an epimorphism φ γ : Pϒ(S γ , P) → Pϒ(S γ , P ∪ {Q}), with kernel N γ the normal subgroup generated by the Dehn twists about symmetric separating simple closed curves on S γ bounding a disc which only contains the punctures labeled by Q and R and no marked points.…”

“…(ii): Assigning to a smooth projective genus 0 curve C endowed with an ordered set of marked points B υ , the hyperelliptic curve C such that C / υ ∼ = C with branch locus B υ , defines a morphism s : PM(S /υ , B υ ) → M(S, υ) (cf. item (h) of Theorem 1.4 in [18]), which, composed with the natural morphism M(S, υ) → M(S /υ , B υ ), is just the natural morphism PM(S /υ , B υ ) → M(S /υ , B υ ).…”