All Questions
Tagged with log-geometry ag.algebraic-geometry
21 questions
1
vote
0
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131
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Can logarithmic blow-ups be constructed étale-locally?
I would like to understand if the logarithmic blow-up of a log scheme can be performed étale locally:
Specifically, suppose that $X^{\dagger}=(X,\mathcal{M}_{X})$ is a fine, saturated log scheme and $...
- 305
4
votes
1
answer
271
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, ...
- 1,104
1
vote
0
answers
83
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-...
- 2,907
3
votes
1
answer
136
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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 $X$...
- 2,907
1
vote
0
answers
115
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 $\...
- 223
2
votes
0
answers
98
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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}: \...
- 223
2
votes
0
answers
113
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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}^{...
- 223
2
votes
1
answer
174
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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
0
answers
243
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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 ...
- 18.7k
7
votes
2
answers
636
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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 $(\...
- 18.7k
3
votes
1
answer
403
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 ...
user138661
4
votes
1
answer
860
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 ...
- 171
7
votes
2
answers
534
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 ...
- 7,962
2
votes
1
answer
180
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 ...
- 7,962
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 ...
- 3,545
6
votes
0
answers
621
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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 ...
- 1,838
2
votes
1
answer
448
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 ...
- 23
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 $...
- 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 ...
- 4,450
6
votes
3
answers
2k
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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 ...
- 16k
10
votes
2
answers
944
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 ...
- 52.7k