The Logarithmic Smoothness of Non-d-Semistable Normal Crossing Schemes

Explore the logarithmic smoothness of non-d-semistable normal crossing schemes in this 54-minute conference talk by Simon Felten from Oxford University. Examine how normal crossing schemes V/ℂ behave when they are d-semistable versus when they are only globally generated, focusing on the construction of log smooth structures over the standard log point S₀. Learn about the challenges of partial log schemes (V, V – Z,ℳ) where the log structure is defined only on open subsets, and discover a new formalism for extending log smooth structures across singular loci Z as sharp lax log structures. Understand how these sharp lax log structures admit charts, are locally isomorphic to spectra of sharp lax log rings, and possess the infinitesimal lifting property in the category of integral sharp lax log schemes. Investigate applications of this formalism to positive and simple toric log Calabi-Yau spaces within the Gross-Siebert program, where the infinitesimal lifting property holds up to codimension 3, corresponding to singularities in codimension 4 for general fibers of toric degenerations. This presentation was delivered at the "New Developments in Singularity Theory" conference at the University of Miami, organized as a joint IMSA & ICMS event with support from the Simons Foundation and National Science Foundation.