How to make log structures

Abstract

We introduce the concept of a viable generically Gorenstein toroidal crossing (ggtc) space $Y$. This generalizes the concept of Gorenstein toroidal crossing scheme, which in turn generalizes that of a simple normal crossing scheme. On such a space $Y$, we define a sheaf $\mathcal{LS}_Y$, intrinsic to $Y$, by means of an explicit construction. Our main theorem establishes a bijection between the set $\operatorname{LS}(Y)$ of isomorphism classes of log structures on $Y$ over the log point $\operatorname{Spec} k^\dagger$ that are compatible with the ggtc structure and the set $\Gamma(Y,\mathcal{LS}_Y^\times)$ of nowhere vanishing global sections of $\mathcal{LS}_Y$. The definition of $\mathcal{LS}_Y$ by explicit construction permits the effective construction of log structures on $Y$; it also enables logarithmic birational geometry, in particular the construction - in some cases - of resolutions of singular log structures. Our work generalizes [GS06], Theorem 3.22, adapting the original proof with techniques from the theory of $2$-groups and local line bundle systems.