Spin curves and Scorza quartics

In the paper (Takagi and Zucconi in “On blow-ups of the quintic del Pezzo 3-fold and varieties of power sums of quartic hypersurfaces”, pp 1–40, preprint, submitted, 2008), we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. In this paper, we show that these quartics coincide with the Scorza quartics of general pairs of trigonal curves and ineffective theta characteristics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartics for any general pairs of curves and ineffective theta characteristics. We also give descriptions of the moduli spaces of trigonal even spin curves.