How to make log structures

Abstract We introduce the concept of a viable generically Gorenstein toroidal crossing (ggtc) space upper Y Y $Y$ . This generalizes the concept of Gorenstein toroidal crossing scheme, which in turn generalizes that of a simple normal crossing scheme. On such a space upper Y Y $Y$ , we define a sheaf script upper L script upper S Subscript upper Y L S Y $\mathcal{LS}_Y$ , intrinsic to upper Y Y $Y$ , by means of an explicit construction. Our main theorem establishes a bijection between the set upper L upper S Subscript k Sub Superscript dagger Baseline left parenthesis upper Y right parenthesis LS k † ⁡ ( Y ) $\operatorname {LS}_{k^\dagger } (Y)$ of isomorphism classes of log structures on upper Y Y $Y$ over the log point upper S p e c k Superscript dagger Spec ⁡ k † $\operatorname {Spec} k^\dagger$ that are compatible with the ggtc structure and the set normal upper Gamma left parenthesis upper Y comma script upper L script upper S Subscript upper Y Superscript times Baseline right parenthesis Γ (