Related papers: Logarithmic Flatness

We introduce the concept of a viable generically Gorenstein toroidal crossing (ggtc) space $Y$. This generalizes the concept of Gorenstein toroidal crossing scheme, which in turn generalizes that of a simple normal crossing scheme. On such…

We study families of matrix Hessenberg schemes in the affine scheme of complex $n\times n$ matrices, each defined over a fixed sheet in the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$. It is well known that such families over the regular…

We compute the rational homology of the moduli stack $\mathcal{M}$ of objects in the derived category of certain smooth complex projective varieties $X$ including toric varieties, flag varieties, curves, surfaces, and some 3- and 4-folds.…

We compare tame actions in the category of schemes with torsors in the category of log schemes endowed with the log flat topology. We prove that actions underlying log flat torsors are tame. Conversely, starting from a tame cover of a…

In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log…

Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to…

Fix a noetherian scheme S. For any flat map f: X->Y of separated essentially-finite-type perfect S-schemes we define a canonical derived-category map c(f):\H(X)->f^!\H(Y), the fundamental class of f, where \H(Z) is the (pre-)Hochschild…

Let $X$ be a fine and saturated log scheme, and let $G$ be a commutative finite flat group scheme over the underlying scheme of $X$. If $G$-torsors for the fppf topology can be thought of as being unramified objects by nature, then…

We extend the notions of Hochschild and cyclic homology to morphisms from algebraic spaces to algebraic stacks. Using this, we obtain generalizations to log schemes in the sense of Fontaine and Illusie of these homology theories.

We investigate flat maps where the source or target is a Noetherian ring, giving necessary and/or sufficient conditions on a ring for such maps to exist. Along the way, we develop some general facts about flat ring maps, and exhibit many…

Given a morphism $X \to S$ of fine log schemes, we develop a geometric description of the sheaves of higher-order differentials $\Omega^n_{X/S}$ for $n > 1$, as well as a definition of the de Rham complex in terms of this description.

We define the logarithmic tautological rings of the moduli spaces of Deligne-Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras…

For each simply connected, simple complex group $G$ we show that the direct sum of all vector bundles of conformal blocks on the moduli stack $\bar{\mathcal{M}}_{g, n}$ of stable marked curves carries the structure of a flat sheaf of…

We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra…

We define log Hochschild co/homology for log schemes that behaves well for simple normal crossing pairs $(X,D)$ or toroidal singularities. We prove a Hochschild-Kostant-Rosenberg isomorphism for log smooth schemes, as well as an equivariant…

In this paper, we investigate the properties of $A$-coherent and $A$-quasi-coherent sheaves within the framework of algebraic geometry over non-algebraically closed fields. We define an $\mathcal{O}_X$-module to be $A$-coherent (resp.…

We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds;…

Enriched curves have been studied over algebraically closed fields by Main\`o ([Mai98]) and recently over general base schemes in [BH19]. In this paper, we study enriched curves from a logarithmic viewpoint: we give a succinct definition of…

We prove that the moduli space of stable logarithmic maps with fixed numerical invariants, from logarithmic curves to a fixed projective target logarithmic scheme with fine and saturated logarithmic structure, is a proper algebraic stack.…

Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical…