All Questions
38 questions
1
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0
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165
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Perfect complexes in a family
Consider a simple normal crossings variety $X=\bigcup_{i=1}^k X_i$ over $\mathbb{C}$ where $X_i$ are smooth projectiv and a flat family $\mathcal{X}\xrightarrow{\pi}\mathbb{A}^1_{\mathbb{C}}$ with $\...
- 341
4
votes
1
answer
723
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Reference request: Schlessinger's Thesis
Does anyone have a copy of Schlessinger's Thesis (not his paper "Functors of Artin Rings")
As other posters have mentioned, this document is cited in Deligne-Rapoport's "Les schemas de ...
- 3,625
4
votes
1
answer
773
views
Schlessinger's thesis
In Deligne-Mumford's "The irreducibility of the space of curves of given genus", the authors use the "Schlessinger's theory",
and refer his "thesis".
Where can I read it?
It seems to be different from ...
- 1,364
15
votes
1
answer
770
views
Cotangent Complex in Analytic Category
I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold ...
- 1,761
5
votes
0
answers
154
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One-parameter family of algebra structures: characterizing trivial deformation as adjoint 2-cocycle being a 2-coboundary
$\DeclareMathOperator\C{\mathbf{C}}$Motivation: this post discusses a simple criterion for a 1-parameter family of $n$-dimensional complex algebras to be a "trivial deformation", i.e., be ...
- 63.9k
5
votes
1
answer
920
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Stacks and Maurer-Cartan elements
One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$.
For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
- 1,609
1
vote
0
answers
84
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Obstruction to deformation of composite morphism (Reference request + question)
Let $f_0:X_0\xrightarrow{g_0}Y_0\xrightarrow{h_0}Z_0/S_0$ be a morphism of smooth projective $S_0$-schemes such that $g_0,h_0$ are flat. Let $S_0\subset S$ be a first-order thickening, and let $X,Y,Z$ ...
- 599
8
votes
1
answer
2k
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Elementary (English) reference for the cotangent complex?
I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack.
I understand that the standard ...
- 83
2
votes
0
answers
131
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versal deformation ring of a p-divisible group with some tensors
I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with ...
- 1,093
26
votes
2
answers
2k
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Strict applications of deformation theory in which to dip one's toe
I hesitate to ask a question like this, but I really have tried finding answers to this question on my own and seemed to come up short. I readily admit this is due to my ignorance of algebraic ...
- 13.5k
2
votes
2
answers
592
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Infinitesimal deformations and moving cycles
The wonderful responses to an earlier question Self-intersection and the normal bundle motivated me to ask the following question:
Let $Y \subset X$ be a subvariety of a variety $X$. Infinitesimal ...
- 3,555
3
votes
1
answer
306
views
Local to global deformation of invertible sheaves
Let $\pi:X \to S$ be a flat, projective morphism, $S$ irreducible. Suppose that for all $s \in S$, the fiber $X_s$ satisfies $h^2(\mathcal{O}_{X_s})=0$. This means in particular that given an ...
- 1,981
1
vote
1
answer
247
views
Infinitesimal deformation of strict transform
Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...
- 2,126
1
vote
1
answer
223
views
Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?
Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
- 2,126
29
votes
1
answer
4k
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Almost Complex Structure approach to Deformation of Compact Complex Manifolds
I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second ...
- 19.3k
1
vote
0
answers
177
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Geometric meaning of residue constraints
$\DeclareMathOperator\Res{Res}$I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/abs/1701.09137) and am having ...
- 425
7
votes
2
answers
1k
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References for the moduli space of complex structures
I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
- 2,816
5
votes
0
answers
141
views
Poincare duality in families of smooth, projective curves
Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \...
- 1,593
3
votes
0
answers
123
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Degeneration of cycle class map
Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...
- 1,593
4
votes
1
answer
2k
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Reference request for an introduction to deformation theory in algebraic geometry
I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
4
votes
1
answer
512
views
Being Cohen-Macaulay open in Hilbert scheme?
Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
- 3,031
3
votes
0
answers
281
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Vanishing of space of first order infinitesimal deformations for irreducible algebraic stack
This question has a few bits, and apologies if some questions are phrased poorly since I am not knowledgeable on the language of stacks or deformation theory.
Suppose $\mathscr{X}$ is an algebraic ...
- 131
3
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0
answers
214
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Representability of Flattening stratification functor
Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
- 1,981
1
vote
1
answer
203
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Assumption of genus at least $2$ for stable curves
In the article "The irreducibility of the space of curves of given genus" by Deligne, Mumford, the definition of stable curves start with the assumption that the genus is at least $2$. Why is this ...
- 1,981
0
votes
1
answer
354
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Reference for Hodge loci on moduli space of principally polarised abelian varieties
Can someone suggest a reference to study Hodge locus, period mappings and period domains on moduli space of principally polarised abelian varieties?
More precisely, consider the moduli space of ...
- 1,981
1
vote
0
answers
126
views
Pull-back of line bundles and field extension
Let $X$ be a smooth, projective variety over a field $K$ of characteristic $0$ (not necessarily algebraically closed) and $L$ an invertible sheaf on $X_{\bar{K}}=X \times_K \mbox{Spec}(\bar{K})$, ...
- 2,032
4
votes
1
answer
200
views
A moduli problem inspired by Stein factorization
Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ ...
- 2,126
1
vote
1
answer
293
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Homological dimension of pure coherent sheaves and specialization
Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{...
- 2,323
2
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0
answers
345
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Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
507
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Fiber of the specialization map of Picard groups
Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
- 2,323
1
vote
0
answers
191
views
Deformation of projective bundles
Let $\pi:\mathcal{X} \to \mathbb{P}^1$ be a flat, projective family of noetherian schemes with generic fiber a smooth, projective variety. Let $p:\mathcal{Y} \to \mathcal{X}$ be another flat, ...
- 2,126
1
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0
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149
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Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
1
vote
0
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381
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On tangent space of relative quot schemes in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic and $f:X \to S$ be a smooth, flat, projective morphism between noetherian $k$-schemes. Assume that $S$ is a non-singular quasi-...
- 503
2
votes
0
answers
213
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Quantities associated to deformed sheaves
I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...
- 155
3
votes
1
answer
468
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Why should we study deformations of perfect complexes
What are the advantanges of studying deformation of perfect complexes over the classical theory of deformation of coherent sheaves? Any references which elaborates on the applications on deformation ...
- 1,593
2
votes
0
answers
177
views
Residual scheme to local complete intersection schemes in the projective space
Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection ...
- 2,126
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 839
5
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0
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372
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Deformation theory with a view toward GW theory and DT theory
I am studying GW theory (and DT theory) in algebraic geometry. I now understand the heuristic "Aut, Def, Obs" argument written in Mirror Symmetry book (by Hori et al.), but it is too hard for me to ...
- 349