Questions tagged [log-geometry]
Log structures, semistable degenerations, log crystalline cohomology, log de Rham cohomology, log smoothness, log Gromov-Witten theory
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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 ...
- 4,400
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votes
0
answers
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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 ...
- 10.7k
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votes
2
answers
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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 ...
- 50.6k
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votes
3
answers
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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 $...
- 647
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votes
1
answer
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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 ...
- 3,465
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votes
2
answers
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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 ...
- 7,882
7
votes
2
answers
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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 $(\...
- 18.1k
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votes
3
answers
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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 ...
- 15.7k
6
votes
1
answer
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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 ...
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votes
0
answers
131
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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:
...
- 1,082
5
votes
0
answers
567
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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 ...
- 1,818
4
votes
1
answer
709
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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 ...
- 171
4
votes
0
answers
374
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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 ...
- 193
3
votes
1
answer
356
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$p$-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 ...
user138661
3
votes
1
answer
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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 ...
- 181
2
votes
1
answer
461
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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, ...
2
votes
1
answer
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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}])...
- 165
2
votes
1
answer
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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 ...
- 7,882
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votes
1
answer
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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 ...
- 23
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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}^{...
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0
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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 ...
- 18.1k
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0
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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 ...