Cubic curves and totally geodesic subvarieties of moduli space
The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. In this paper the authors show that the flex locus provides the first example of a primitive totally geodesic subvariety of moduli space and the gothic locus provides new \(\mathrm{SL}_2(\mathbb R)\)-invariant varieties in Teichmüller dynamics. A number of interesting properties of these loci from the viewpoints of projective geometry and flat surfaces are discussed.
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