Abstract

The Hilbert scheme of a projective variety X is the moduli space of closed subschemes of X. In this talk we discuss a version of the Hilbert scheme for a pair (X,D) with D a (reasonable) divisor on X. This logarithmic Hilbert scheme is the most intuitive example of the logarithmic Quot scheme. There is a closely related tropical moduli problem called the stack of tropical supports. This tropical problem is reminiscent of a tropical version of a Hilbert scheme. As motivation, note hard algebraic geometry problems can be studied by degenerating to simpler situations. Logarithmic geometry provides a suite of tools to study such degenerations. A long term hope is to study other moduli spaces of coherent sheaves with logarithmic geometry.