All Questions
Tagged with ag.algebraic-geometry deformation-theory
26 questions
64
votes
5
answers
9k
views
Intuition about the cotangent complex?
Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
- 12.3k
32
votes
4
answers
2k
views
Clifford algebras as deformations of exterior algebras
$\def\Cl{\mathcal C\ell}
\def\CL{\boldsymbol{\mathscr{C\kern-.1eml}}(\mathbb R)}$
I'm not an expert in neither of the fields I'm touching, so don't be too rude with me :-) here's my question.
A well ...
- 13.6k
11
votes
2
answers
780
views
Deformations of a blowup
Let $S$ be a smooth projective surface over $\mathbb{C}$. (I guess this can be more general—higher dimension, other ground fields, non-projective, maybe even singular?—and I'dd like to hear that.) Let ...
- 5,504
5
votes
2
answers
984
views
Injectivity under flat base change of the Picard group on smooth projective curves
Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...
- 503
2
votes
1
answer
205
views
Deformation of isolated singularities and non zero divisors
Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...
29
votes
1
answer
4k
views
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
19
votes
4
answers
2k
views
Example of a smooth morphism where you can't lift a map from a nilpotent thickening?
Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. étale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...
17
votes
2
answers
1k
views
What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
- 10.5k
14
votes
2
answers
2k
views
"Spec" of graded rings?
From the discussion at Hochschild cohomology and A-infinity deformations, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have ...
- 21k
13
votes
1
answer
2k
views
What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?
Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
- 6,069
12
votes
1
answer
1k
views
Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 2
This is a sequel to the question Accumulation of algebraic subvarieties: Near one subvariety there are many others (?) .
Let $Y$ be some projective variety, over $\mathbb{C}$. Let $X\subset Y$ be ...
- 21.3k
11
votes
1
answer
930
views
Deformations of smooth projective hypersurfaces and the Jacobian ring
It is a well-known result of Griffiths that the pieces of Hodge filtration of a smooth hypersurface $X:= (f=0)$ of degree $d$ in $\mathbb{P}^{n}$ are isomorphic to graded pieces of the Jacobian ring ...
- 637
10
votes
2
answers
3k
views
Kodaira-Spencer map in a concrete instance
Let $\pi:X_{\epsilon} \rightarrow \Delta$ be a family of (say smooth) projective plane curves parametrized by $\Delta:=\operatorname{Spec}(k[\epsilon])$, and let $X=X_0$ be the closed fiber.
Suppose ...
- 23.3k
10
votes
2
answers
1k
views
Some examples of $\mathbb Q$-Gorenstein smoothing
I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition.
Definition. For a normal projective surface $X$ with quotient ...
- 641
9
votes
2
answers
2k
views
What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?
Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
- 10.5k
7
votes
1
answer
1k
views
Can the homological dimension of a coherent sheaf explode along a formal deformation? (is the resolution property hereditary for formal deformations?)
Let $X_0$ be a locally noetherian scheme and $\mathcal{F}_0$ a coherent $\mathcal{O}_{X_0}$-module. Let $C$ be an artin ring with residue field $k$ and let $X \to Spec C$ be a (flat) deformation of $...
- 7,789
5
votes
0
answers
189
views
Extension of holomorphic maps to smooth family of holomorphic maps
Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
- 1,409
5
votes
2
answers
1k
views
Exercise 1.1.(c) in Hartshorne's Deformation Theory
Exercise 1.1.(c) in Hartshorne's Deformation Theory:
Over an algebraically closed field $k$, we define a curve in $\mathbb P^2_k$ to be the closed subscheme, defined by a homogeneous polynomial $f(...
- 159
5
votes
0
answers
154
views
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
307
views
Infinitesimal deformations of fake projective planes (or ball quotients)
This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By ...
- 66.3k
4
votes
5
answers
582
views
Reference Request: Deformations of a map bijective to global sections of the pullback of the tangent sheaf
I have been trying to learn some deformation theory, and came across the following in a paper:
The first order deformations of a morphism of smooth curves $f:X\rightarrow Y$ is in bijection with $H^0(...
- 801
2
votes
1
answer
735
views
A question on nested Hilbert scheme
Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in $\...
- 2,022
2
votes
2
answers
575
views
Infinitesimal deformations of a fibration
Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...
user68440
2
votes
1
answer
474
views
Deformations of pointed stable maps with "curve held rigid" or "preserving the dual graph"
I am reading the "Notes on stable maps and quantum cohomology" by Fulton and Pandharipande and I got stuck in the proof of the Theorem 2, p.27. The authors consider the space $Def(\mu)$ of first order ...
- 203
2
votes
1
answer
265
views
Question about automorphism functor in Sernesi's "Deformations of algebraic schemes"
Let $X$ denote an algebraic scheme over $\operatorname{Spec} k$ such that its deformation functor $\operatorname{Def}_X$ has a semi-universal couple $(R,u)$, where $R$ is an Artinian $k$-algebra and $...
- 749
1
vote
0
answers
293
views
Specialization map and fibration
Let $\pi:X \to \Delta$ be a proper, surjective, flat morphism (here $\Delta$ is the unit disc), smooth over $\Delta \backslash \{0\}$ and possibly singular central fiber. There is a fibrewise ...
- 2,323