Moduli of quartic surfaces and automorphic forms on symmetric domains of type IV

Contrary to dimensions $1$ and $2$, almost nothing was known about the structure of algebras of automorphic forms on multidimensional symmetric domains of type IV. The only such result was obtained by J. Igusa (1962), who proved that some algebra of automorphic forms on the $3$-dimensional symmetric domain of type IV is free, and found the degrees of its generators. In 2010, the speaker managed to obtain analogous results in dimensions $4$, $5$, $6$, $7$, making use of the interpretation of the projective spectra of the considered algebras of automorphic forms as the moduli varieties of some classes of quartic surfaces. These results imply, in particular, that the corresponding arithmetic groups are generated by reflections. On the other hand, the speaker recently proved an old conjecture of O. V. Shvartsman (1981) on singularities at infinity of arithmetic quotients of symmetric domains of type IV, which implies that the algebra of automorphic forms on the $n$-dimensional symmetric domain of type IV with respect to some arithmetic group may be free only if $n\leqslant 10$. Moreover, it seems that there are only finitely many arithmetic groups with this property.