2023/12/21 by Corti, Alessio, Ruddat, Helge
#14A21 #14B07 #14D23 #14E15 #14J33 #14J45 #14M25 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2312.13867
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 LSY, intrinsic to Y. Our main theorem states that the set of nowhere vanishing sections Γ(Y,LSY^×) is canonically bijective to the set of isomorphism classes of log structures on Y over k^† compatible with the gtc structure. The definition of LSY 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.