Skip to main content

Questions tagged [log-geometry]

Log structures, semistable degenerations, log crystalline cohomology, log de Rham cohomology, log smoothness, log Gromov-Witten theory

Filter by
Sorted by
Tagged with
4 votes
1 answer
256 views

Intersection complex of genus-zero curves?

I would like to have a very explicit description of $\bar M_{0, n}$, especially its boundary divisors and how they intersect. All I can do in my construction is add divisors and blow up at strata, ...
Leo Herr's user avatar
  • 1,084
1 vote
0 answers
74 views

Are coherent modules with integrable log-connections locally free?

Let $X$ be a smooth Noetherian scheme over a field $K$. It is known that every coherent module with integrable connection on $X$ is locally free. Is the same true for coherent modules with log-...
kindasorta's user avatar
  • 2,085
3 votes
1 answer
119 views

Reference request for log-differential forms

I read in a paper of Kato about log-differential forms, that if $X$ is a smooth locally Noetherian log-scheme, and $D$ is a reduced normal crossing divisor, then there is a definition of a sheaf on $X$...
kindasorta's user avatar
  • 2,085
1 vote
0 answers
99 views

Cokernel of map of dual of sheaves of differentials/deformations

Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
Matthias's user avatar
  • 203
2 votes
0 answers
90 views

Tangent Space of Moduli of Log-Smooth Curves

We consider an algebraically closed field $\underline{k}$ and all constructions that we will consider are over this field. It is well known that for each relative nodal curve $\underline{f}: \...
Matthias's user avatar
  • 203
2 votes
0 answers
111 views

Equivalence between $\bar{\mathcal{M}}_{g,n}$ and ${\mathcal{M}}_{g,n}^{logbas}$

It is a classical result of the theory of the moduli of curves, that the stack $\bar{\mathcal{M}}_{g,n}$ of nodal curves with log-structure coming from the boundary divisor, and ${\mathcal{M}}_{g,n}^{...
Matthias's user avatar
  • 203
2 votes
1 answer
165 views

Understanding the picture of monoidal space

Ogus in his slides https://math.berkeley.edu/~ogus/preprints/colloqhandout.pdf presents the following picture of a monoidal space $\operatorname{Spec}(\mathbb{N} \longrightarrow \mathbb{C}[\mathbb{N}])...
Nicholas S's user avatar
3 votes
1 answer
282 views

Reference request: Kummer étale topology and tame topology

In Theorem 7.6 of Illusie - An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology it is stated that if $X$ is a log regular fs log scheme and $U$ is the open ...
user197402's user avatar
2 votes
0 answers
235 views

Terminology in log geometry

A log scheme consists of a scheme $X$, a sheaf of monoids $M_X$ on $X$, and a map $\alpha:M_X\to\mathcal O_X$ with the property that $\alpha^{-1}(\mathcal O_X^\times)\to\mathcal O_X^\times$ is an ...
John Pardon's user avatar
  • 18.4k
7 votes
2 answers
603 views

Is $(x^2y,xy^2)$ log smooth?

Consider the map $$f:\mathbb C^2\to\mathbb C^2$$ $$(x,y)\mapsto(x^2y,xy^2)$$ We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix $(\...
John Pardon's user avatar
  • 18.4k
5 votes
0 answers
138 views

algebraic de Rham cohomology of toric varieties (reference request)

I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled: ...
Somatic Custard's user avatar
6 votes
1 answer
290 views

An unpublished note by Bloch-Kato on p-divisible groups and Dieudonné crystals

I wonder if anyone could find the following unpublished paper of Bloch-Kato: Spencer Bloch and Kazuya Kato, $p$-divisible groups and Dieudonné crystals, unpublished. A similar question is here ...
Mojo's user avatar
  • 85
3 votes
1 answer
374 views

$p$-adic Kato--Nakayama space

Given a log scheme over $\mathbb{C}$ whose underlying scheme is locally of finite type, you can associate to it a ringed space called the Kato--Nakayama space. Is there a $p$-adic analogue of this ...
user avatar
4 votes
1 answer
793 views

Properties of log smooth schemes

Let $k$ be a field and $M$ be sharp monoid (with no invertible element) consider the log point $\eta_M=(\operatorname{Spec}(k), M)$. Let $X$ be a fine saturated scheme over $\eta_M$ such that the ...
ABC's user avatar
  • 171
7 votes
2 answers
503 views

A log structure on the moduli space of curves

Let $M_{g, n}$ be the moduli space of curves of genus $g$ with $n$ marked points. Let $M_{g, \vec{n}}$ be the moduli space of marked curves with a choice of a (possibly zero) tangent vector at each ...
Dmitry Vaintrob's user avatar
2 votes
1 answer
177 views

Locally toric resolutions of compactifications

Suppose $U$ is a smooth, open $n$-dimensional variety over $\mathbb{C}.$ Say $X, X'$ are two proper normal-crossings compactifications of $U$. Call a map $m: X'\to X$ a modification if it is an ...
Dmitry Vaintrob's user avatar
4 votes
0 answers
384 views

An unpublished note by Spencer Bloch and Kazuya Kato

I am looking for an unpublished note by Spencer Bloch and Kazuya Kato, p-divisible groups and Dieudonné crystals. This note is always cited as Spencer Bloch and Kazuya Kato, p-divisible groups and ...
Mayday's user avatar
  • 193
8 votes
1 answer
1k views

When is a map from a logarithmic tangent bundle to a normal bundle surjective?

Suppose that $X$ is an algebraic variety with divisor $D$ such that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a regular ...
Simon Wadsley's user avatar
5 votes
0 answers
596 views

Semistable reduction and log structures

I have been reading Hyodo-Kato's paper on log-crystalline cohomology, and there is one statement there that has been troubling me. To explain this, suppose we have a perfect field $k$ of ...
ChrisLazda's user avatar
  • 1,828
2 votes
1 answer
433 views

trivialities on log-structures

I would like to understand some trivialities about log-structures. Given a log-scheme $(X,M_X)$ the log-structure $M_X$ is defined via push-out. Are there stupid examples in which this push-out is ...
ketth's user avatar
  • 23
2 votes
1 answer
464 views

Minimal semistable model for K3-surfaces.

I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference. Also, if we have a semistable K3 surface with a log structure, ...
Rogelio Yoyontzin's user avatar
1 vote
0 answers
233 views

Moduli Space of an Algebraic K3 surface with singularities.

Suppose that $X$ is an algebraic K3 surface (say polarized). If the singular divisor of $X$ is normal crossing... Do we have a moduli space parametrizing such $K3$ surfaces? If yes do we have a ...
Rogelio Yoyontzin's user avatar
8 votes
3 answers
1k views

relation between toric geometry and log geometry

Hello, I'm trying to understand the relation between the points of view of log geometry (monoids) and toric geometry (fans). Suppose that $k$ is a field and $P$ is a finitely generated monoid. Then $...
unknown's user avatar
  • 647
15 votes
4 answers
5k views

References for logarithmic geometry

Hi everyone, I'm looking for a systematical introduction to (or treatment of) logarithmic structures on schemes. I am reading Kato's article ("Logarithmic structures of Fontaine-Illusie") at the ...
Lars's user avatar
  • 4,420
6 votes
3 answers
1k views

What are Log Stacks

So, I've been running in both stacky circles and logarithmic circles and I've been wondering: is there a definition of log stack that is "useful"? I can imagine two such definitions: 1) A log stack ...
Charles Siegel's user avatar
13 votes
0 answers
821 views

Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk? I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...
Thomas Riepe's user avatar
  • 10.8k
9 votes
2 answers
921 views

Logarithmic structures on moduli of elliptic curves over Z

I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...
Tyler Lawson's user avatar
  • 51.9k