This paper studies the gradient flow lines for the \(L^2\) norm square of the Higgs field defined on the moduli space of semistable rank 2 Higgs bundles twisted by a line bundle of positive degree over a compact Riemann surface X. The main result is that these spaces of flow lines have an algebro-geometric classification in terms of secant varieties for different embeddings of X into the projectivisation of the negative eigenspace of the Hessian at a critical point. The Morse-theoretic compactification of spaces of flow lines given by adding broken flow lines then has a natural algebraic interpretation via a projection to Bertram’s resolution of secant varieties.
