Infinitesimal invariants of genus 4 curves and the Griffiths-Pirola cycle

We give an explicit algebraic description for the infinitesimal invariants associated to the Griffiths-Pirola cycle on the universal genus 4 curve and its self intersection.

We apply this to refine a result of Griffiths which recovers from these invariants the equations for the general genus 4 curve, and give a new proof (only in genus 4) of a nonvanishing result of Green and Griffiths.