Segre quartic surfaces and minitwistor spaces (Q2140551)

Twistor spaces are complex manifolds containing certain families of complex curves which are parametrized by manifolds that carry interesting geometric structures. Indeed, twistor spaces were originally invented as a tool to study Riemannian four-manifolds carrying a self-dual Einstein metric. In this case, the twistor space is three-dimensional and the self-dual Einstein four-manifold is recovered as the parameter space of a family of smooth, rational curves with normal bundle \(\mathcal O(1)\oplus \mathcal O(1)\) satisfying a reality condition. \textit{N.~J.~Hitchin} [Lect. Notes Math. 970, 79--99 (1982; Zbl 0507.53025)] uncovered a similar twistor correspondence between certain projective surfaces containing a smooth, rational curve with self-intersection number two. These surfaces are known as minitwistor spaces and it is known that the deformations of the rational curve, which is called a minitwistor line, are parametrized by a Zariski-open subset \(W_0\) of a smooth, projective three-fold \(W\), carrying a so-called (complex) Einstein-Weyl structure. This theory, in turn, was generalized by the author and \textit{F.~Nakata} [Ann.~Global Anal.~Geom.~39, No.~3, 293--323 (2011; Zbl 1222.53053)] to allow the minitwistor line to have \(g\) nodes and self-intersection \(2+2g\), allowing for more examples. The minitwistor line is a degeneration of smooth curves of genus \(g\), hence \(g\) is called the genus of the minitwistor space. In this article, the author studies the case \(g=1\), identifying the projective three-fold \(W\) as the (projective) dual of the minitwistor space. This is then used to show that essential minitwistor spaces of genus one are precisely the so-called Segre quartic surfaces. This is a classically known class of surfaces, which arise as complete intersections of two quadrics in \(\mathbb C\mathrm P^4\). The author uses this description to study minitwistor spaces and their dual varieties, making use of the classification of Segre surfaces. Particular attention is paid to the complement of the Weyl-Einstein space \(W_0\) inside \(W\), whose two-dimensional components the author calls divisors at infinity.