On blow-ups of the quintic del Pezzo 3-fold and varieties of power sums of quartic hypersurfaces

TL;DR

This paper constructs new subvarieties within the varieties of power sums for specific quartic hypersurfaces, generalizing Mukai's description of certain Fano threefolds and linking to the main conjecture of Dolgachev and Kanev.

Contribution

It introduces new subvarieties in the varieties of power sums for quartic hypersurfaces and connects these to the Scorza quartics and the conjecture of Dolgachev and Kanev.

Findings

New subvarieties in varieties of power sums for quartic hypersurfaces

Generalization of Mukai's description of Fano threefolds

Proof of the main conjecture of Dolgachev and Kanev

Abstract

We construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. This provides a generalization of Mukai's description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. In fact, in our second paper, we show that these quartics are exactly the Scorza quartics associated to general pairs of trigonal curves and ineffective theta characteristics and this enables us to prove there the main cojecture of Dolgachev and Kanev.