Git Versus BB Compactification for the Moduli Space of Quartic Surfaces

This is a report on joint work with Radu Laza. The period map from the GIT moduli space of quartic surfaces to the Baily-Borel compactification of the period space is birational but far from regular. New birational models of locally symmetric varieties of Type IV have been introduced by Looijenga, in order to study similar problems. Looijenga's construction does not succeed in “explaining” the period map for quartic surfaces. We discovered that one can (conjecturally) reconcile Looijenga's philosophy with the phenomenology of quartic surfaces, provided one takes into account suitable Borcherd relations between divisor classes on relevant locally symmetric varieties. We work with a tower of locally symmetric varieties, in particular our results should also “explain” the period map for double EPW sextics.