Segre surfaces are quartic surfaces in CP^4
and they are anti-canonical models of smooth del-Pezzo surfaces of degree four.
We characterize these surfaces as minitwistor spaces by a numerical invariant.
By a kind of Penrose correspondence, this means that the smooth locus of
the dual varieties of Segre surfaces admit an Einstein-Weyl structure.
We investigate these dual varieties in detail,
possibly with some emphasize on the locus which parameterizes cuspidal rational curves.
This is partially joint work with my student Ayato Minagawa.
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