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Segre surfaces in the title mean quartic surfaces in $\mathbb{CP}^4$ whichare the images of weak del Pezzo surfaces of degree four under theanti-canonical map. We first show that minimal minitwistor spaces with genusone are exactly Segre quartic surfaces. By a kind of Penrose correspondence,Zariski open subsets of the projective dual varieties of these surfaces admitEinstein-Weyl structure. We investigate structures of these dual varieties indetail. In particular, we determine the degrees of these varieties (namely theclasses of the Segre surfaces), as well as structure of several components ofthe boundary divisors which are the complements of the Einstein-Weyl spaces inthe projective dual varieties.

Segre quartic surfaces and minitwistor spaces