Compactifying moduli of hyperelliptic curves

Hyperelliptic curves are algebraic curves that look like two sheets over a line, branching at 2g+2 points. The moduli space H_g collects all such curves of a fixed genus g. To study limits, mathematicians add boundary objects, making the space compact. Two standard compactifications exist: one uses stable curves of genus zero with 2g+2 marked points (N), the other uses polynomial forms of degree 2g+2 up to linear changes of variables (B). This paper constructs a third compactification J_g, built from configurations of lines: take the 2g+2 branch points on a conic and draw all lines through pairs of them. Closing up all such configurations and taking a GIT quotient, which identifies configurations that differ only by a change of coordinates, gives J_g. The main technical result is a recovery theorem: under a natural stability condition, the configuration of lines determines the original conic and its marked points uniquely. This generalizes the classical fact that a smooth plane quartic can be recovered from its bitangents. Using this, the paper defines maps from N to J and from J to B, and proves their composition is exactly the previously known comparison map between N and B. The boundary points of J are configurations with non-reduced lines, and they are interpreted as limits of higher spin curves of order g-1; for g=3 they are the 28 odd theta characteristics. The work is purely mathematical, with no computational or experimental component.