Self-dual metrics and twenty-eight bitangents

We consider self-dual metrics on 3CP^2 of positive scalar curvature admitting a non-trivial Killing field, but which is not conformally isometric to LeBrun's metrics. Firstly, we determine defining equations of the twistor spaces of such self-dual metrics. Next we prove that conversely, the complex threefolds defined by the equations always become twistor spaces of self-dual metrics on 3CP^2 of the above kind. As a corollary, we determine a global structure of the moduli spaces of these self-dual metrics; namely we show that the moduli space is non-empty and isomorphic to R^3/G, where G is an involution of R^3 having one-dimensional fixed locus. Combined with works of LeBrun, this settles a moduli problem of self-dual metrics on 3CP^2 of positive scalar curvature admitting a non-trivial Killing field. In our proof, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.