Toroidal Embeddings I

Newton polyhedra (algebra and geometry)

“…They play the same role as the projective compactification for general varieties of fixed degrees. All these compactifications are what is known as toric varieties [13].…”

Skeletons and Fans of Logarithmic Structures

“…This is a reformulation of the polyhedral cone complexes of [KKMSD73] which realizes them within the category of monoidal spaces. In [Uli13], one further generalizes the construction of the associated Kato fan to logarithmic structures without monodromy.…”

“…Toric varieties, toroidal embeddings and logarithmic structures. Toric varieties were introduced in [Dem70] and studied in many sources, see for instance [KKMSD73,Oda78,Dan78,Oda88,Ful93,CLS11]. They are the quintessence of combinatorial algebraic geometry: there is a category of combinatorial objects called fans, which is equivalent to the category of toric varieties with torus-equivariant morphisms between them.…”

“…In [KKMSD73] some of this picture was generalized to toroidal embeddings, especially toroidal embeddings without self intersections, and their corresponding polyhedral cone complexes (Section 2.7). Following Kato [Kat94] we argue that the correct generality is that of fine and saturated logarithmic structures.…”

On the Jung method in positive characteristic

“…The branch locus is a plane curve C. By embedded resolution of plane curves, one reduces to the case that the only singular points of C are normal crossings. The The singularities of ,S' are toric [19], and a resolution of singularities can be described explicitly from some combinatorial data associated with the set of exponents of the Mj's. This fact is the main feature in the Jung method.…”