Rational and irrational singular quartic threefolds

Burkhardt and Igusa quartics admit a faithful action of the symmetric group of degree 6.
There are other quartic threefolds with this property. All of them are singular.
Beauville proved that all but four of them are irrational. Burkhardt and Igusa quartics
are known to be rational.
Two constructions of Todd imply the rationality of the remaining two quartic threefolds.
In this talk, I will give an alternative proof of both these (irrationality and rationality) results.
This proof is based on explicit small resolutions of so-called Coble fourfold.
This fourfold is the double cover of the four-dimensional projective space branched over
Igusa quartic.
This is a joint work with Sasha Kuznetsov and Costya Shramov from Moscow.