The Segre surfaces of the title are by definition those quartic surfaces in CP₄ that arise as images of weak del Pezzo surfaces of degree four under the anti-canonical map. We first show that under some minimality condition, minitwistor spaces of genus one are exactly Segre quartic surfaces. By a kind of Penrose correspondence, Zariski open subsets of the projective dual varieties of these surfaces admit Einstein-Weyl structure. We investigate structures of these dual varieties in detail. In particular, we determine the degrees of these varieties (namely the classes of the Segre surfaces), as well as structure of several components of the divisors at infinity, which are the complements of the Einstein-Weyl spaces in the projective dual varieties.

The author has been partially supported by JSPS KAKENHI Grant 16H03932.