How to make log structures

Abstract

We introduce the concept of a viable generically toroidal crossing (gtc) Deligne--Mumford stack $Y$. This generalizes the concept of Gorenstein toroidal crossing space, which in turn generalizes that of a simple normal crossing scheme. On such a space $Y$, we define by explicit construction a natural sheaf $\mathcal{LS}_Y$, intrinsic to $Y$. Our main theorem states that the set of nowhere vanishing sections $Γ(Y,\mathcal{LS}_Y^\times)$ is canonically bijective to the set of isomorphism classes of log structures on $Y$ over $k^\dagger$ compatible with the gtc structure. 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 Theorem 3.22 in GS06 and our proof follows closely the proof of that theorem as given in GS06.

Figures (7)

• Figure 4.1: The stalks of the sheaf $\mathcal{M}=\mathcal{P}^{\text{gp}}/\mathbf{1}$ at various points of $T$ for the normal crossing surface $xyz=0$.
• Figure 4.2: The stalks of the relation sheaf $\mathcal{R}$ at various points of $T$ for the normal crossing surface $xyz=0$.
• Figure 4.3: $\overline{U}_{\tau}\subset\overline{U}_{\tau'}$ for $\tau^\prime \leq \tau$
• Figure 5.1: The sublattice $M\subset\mathbb{Z}^2$ for $r=3$.
• Figure 5.2: The moment polyhedral complex of the surface $X$

Theorems & Definitions (56)

• Definition 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Example 2.4
• Example 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Remark 2.9