From hyperelliptic to superelliptic curves


The book will hopefully be finished by Fall 2018. I am looking for a publisher

Part 1:
  • Curves and their automorphisms
    • Curves and their automorphisms
    • Automorphism groups of hyperelliptic curves
    • Superelliptic curves over characteristic 0 fields
    • Superelliptic curves over positive characteristic fields
  • Superelliptic curves and binary forms
    • An introduction to binary forms
    • Transvections and root differences
    • Ring of invariants (Gordan, Noether, Hilbert)
    • Computational invariant theory
  • Weierstrass points
    • Linear systems, inflection points, Wronskian
    • Weierstrass points
    • Hurwitz theorem
    • Weierstrass points of superelliptic curves
  • Superelliptic loci in the moduli space of curves
    • Moduli space of covers and Hurwitz spaces
    • Signatures of automorphism groups
    • Superelliptic loci in M_g
  • Equations of curves with fixed automorphism group
    • Equations of hyperelliptic curves with fixed automorphism group
    • Equations of superelliptic curves with fixed automorphism group
    • Equations over fields of positive characteristic
  • Theta functions
    • Theta functions of curves
    • Frobenious formula
    • Thomae's formula for cyclic coverings
    • Thetanulls of superelliptic curves
  • Superelliptic Jacobians
    • Jacobians of curves
    • Hyperelliptic Jacobians
    • Superelliptic Jacobians
    • Decomposable Jacobians
Part 2:  Minimal models

  • On the field of moduli of curves
    • Field of moduli versus field of definition
    • Hyperelliptic curves over a minimal field of definition
    • Equation of superelliptic curves over a minimal field of definition
    • Equation of superelliptic curves with extra automorphisms
  • Neron models
    • Neron models of curves
    • Neron models for hyperelliptic curves
    • neron models for superelliptic curves
  • Reduction theory of binary forms
    • An introduction to reduction theory
    • Reduction of quadratics
    • Julia quadratic and reduction of cubics
    • Higher degree binary forms
  • Weighted projective spaces
    • An introduction to weighted projective spaces
    • Weighted moduli space of binary forms
    • Heights on weighted projective spaces
  • Twists of curves
    • Twists of algebraic curves
    • Hyperelliptic and superelliptic twists
    • Twists of bounded height
Part 3:  Rational points