All Questions
Tagged with deformation-theory ag.algebraic-geometry
453 questions
1
vote
0
answers
101
views
On descending a section of a morphism between schemes from formal completion to étale local
Here's the case, which arises from the context of doing infinitesimal deformation. Given a DVR $(S,\mathfrak{m},\kappa)$ we have the completion $\hat{S}$ with respect to $\mathfrak{m}$. Say we have a ...
- 11
2
votes
0
answers
129
views
Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?
$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
- 9,019
3
votes
0
answers
156
views
A possible application of deformation theory?
Let $f : \mathbb{R}^{n} \to \mathbb{R}$ be a real-valued polynomial function. Consider the family of real algebraic sets:
$$
V_c = f^{-1}(c), \quad c \in (-1,1).
$$
I am interested in determining how ...
- 357
4
votes
1
answer
243
views
On the degeneration of the elliptic surface $E(n)$
The following matter should be widely known (if true). I am sorry for my ignorance!
For the natural $n$, let $E(n)$ be the corresponding elliptic surface.
In the analytic world, there exists a well-...
- 153
1
vote
0
answers
165
views
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
3
votes
0
answers
186
views
$H^2(X,T_X)=0$ implies the Frölicher spectral sequence degenerates at $E_1$?
Let $X$ be a compact complex manifold, if $X$ satisfies $H^2(X,T_X)=0$, then it is well-known that the Kuranishi space of $X$ is smooth by Kodaira and Spencer's deformation theory. On the other hand, ...
- 471
1
vote
0
answers
96
views
Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
- 178
4
votes
1
answer
723
views
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
2
votes
1
answer
208
views
Infinitesimal neighborhood and Ext group
$\DeclareMathOperator\Ext{Ext}$Let $X$ be a smooth projective complex variety and $\iota\colon Y\subset X$ be a smooth closed subvariety. It is well-known that there is a spectral sequence
$$E_2^{p,q}=...
- 389
1
vote
0
answers
132
views
Extension of MMP from the central fiber to some neighborhood
I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 )
There is a theorem about the extension of MMP step when the central fiber has ...
- 225
1
vote
0
answers
109
views
One question about Manetti surface
I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof.
Theorem 5.2 states that fixed a ...
- 41
2
votes
0
answers
139
views
Question about pro-representable Automorphism functor in Sernesi's "Deformations of algebraic schemes" with trivial relative Tangent space
Let $k$ be an algebraically closed field of arbitrary characteristic, $R$ a complete local noetherian, but non Artinian (see #Edit below for justification) $k$-algebra with residue field $k$ and $S$ a ...
- 6,018
2
votes
0
answers
129
views
Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
- 279
2
votes
1
answer
363
views
Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?
To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
- 21
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
5
votes
0
answers
284
views
Formal neighborhood of stable curves
For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “...
- 834
3
votes
0
answers
171
views
Grothendieck-Messing in characteristic 0?
Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example).
If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
- 261
2
votes
0
answers
354
views
Square-zero extensions mod $p^n$
$\DeclareMathOperator\LMod{LMod}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Sp{Sp}$A square-zero extensions of rings is, conceptually, a map of rings $R \to A$ such that any two elements in the ...
- 68
2
votes
0
answers
100
views
Deformations of invertible sheaves admitting global sections
We follow Sernesi's treatment of algebraic deformations, working over the complex numbers.
Given a pair $(X,L)$ consisting of a compact complex manifold $X$ and an invertible sheaf $L$ on $X$, we ...
- 518
2
votes
0
answers
110
views
Invariance of plurigenera: singular surface case
The invariance of plurigenera is one most central problems in deformation theory. I mainly concern about the (singular) surface setting in this post since
Iitaka had proved that the deformation ...
- 295
3
votes
1
answer
167
views
semiample of canonical bundle in a smooth family (Campana's proof)
The following lemma is due to Campana, The class $\mathcal C$ is not stable by small deformations
Let $\mathcal X\rightarrow \Delta$ be a smooth family, if $K_{X_0}$ is nef and big, then so is every ...
- 295
3
votes
0
answers
132
views
Obstruction to the existence of a deformation of a subvariety compatible with the given deformation of a variety
Let $X$ be a smooth projective variety over a field $k$ of characteristic 0,
and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$
where $I$ is an ideal such that $I^2=0$.
Let $\frak X$ be a ...
- 14.1k
5
votes
0
answers
133
views
What classifies deformations of group schemes (or Hopf algebras)?
The cotangent complex of a scheme classifies its deformations.
That is, if $X$ is a scheme over a field $k$ (with conditions?) and $\mathbf{T}^*_X\in D^b(\text{QCoh}(X))$ its cotangent complex, the ...
- 5,701
2
votes
0
answers
107
views
Deformation of complex manifolds that admit reduction modulo $p$
Let $(M,B,\omega)$ be a complex analytic family of compact (projective non singular) complex manifolds, where $B \subset \mathbb{C}^{m}$ is some domain. Lets consider a subclass of such manifolds $\{...
- 331
3
votes
1
answer
254
views
Degeneration of curves in smooth families
Heuristically, I want to know, given a smooth, projective morphism from a scheme to a discrete valuation ring, if the generic fiber can be 'covered' by a family of geometrically integral curves, is it ...
- 2,323
1
vote
0
answers
93
views
Deform a certain $\mathbb{P}^2$ in $\mathbb{G}(1,4)$
Let $C\subset\mathbb{P}^4$ a rational normal curve. Then the variety of secant lines in $\mathbb{G}(1,4)$ is isomorphic to the second symmetric product of $C$, hence a $\mathbb{P}^2$. Is there a small ...
- 3,031
1
vote
0
answers
240
views
Unexpected holomorphic tubular neighborhood
While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular ...
- 1,761
3
votes
1
answer
326
views
Segre embedding and intersections by hyperplanes
Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...
- 2,323
5
votes
1
answer
439
views
Does the Jacobian functor respect deformations?
I am trying to understand the relationship between deformations of curves and deformations of their Jacobians and would greatly appreciate a sanity check. Let $C_0$ be a smooth projective curve over a ...
- 3,539
2
votes
0
answers
99
views
Reducedness assumption on $X/S$ in Sernesi's Deformations of Algebraic Schemes
I have a question about the proof of a result from
Edoardo Sernesi's Deformations of Algebraic Schemes:
Theorem 1.1.10. Let $X \to S$ be a morphism of finite type of
algebraic schemes and $\mathcal{I}...
- 619
3
votes
1
answer
206
views
Lower bound for the dimension of the space of deformations $\mathrm{Defor}(f : X \to Y)$ in relative setting
Let $f: X \to Y$ a morphism between smooth varieties
over alg. closed field of characteristic zero. It is known that the deformation theory
in relative setting of $f$ is encoded in the cohomology of ...
- 619
3
votes
1
answer
316
views
Exercise 1.5.8 from Robin Hartshorne's Deformation Theory
I don't know how to solve part (a) of exercise 1.5.8 in Robin Hartshorne's book Deformation Theory 1.5.8 (page 42):
Consider the Hilbert scheme of zero-dimensional closed subschemes
of $\mathbb{P}^...
- 619
1
vote
0
answers
133
views
Curve without infinitesimal automorphism has no deformation with automorphism
$\DeclareMathOperator\Spec{Spec}$From studying the classical literature on deformation theory, I have the impression that if the identity morphism of a curve $C$ over an algebraically closed field $k$ ...
- 223
1
vote
0
answers
176
views
Obstruction map for holomorphic line bundle $\operatorname{Ob}_L:H^1(X,\mathcal O)\to H^2(X,\mathcal O)$
$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the ...
- 471
1
vote
0
answers
257
views
When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?
Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$.
Let $L_t$ be a holomorphic line bundle over $X_t$, then its ...
- 471
3
votes
0
answers
164
views
Linear deformations of a morphism between stacks
Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$?
In ...
- 163
1
vote
0
answers
340
views
Deformation theory of stacks and the tangent complex
On a smooth stack X one can construct the two-term tangent complex $T_X \in D(X)$, as in Sam Raskin's notes (https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf).
I ...
- 163
5
votes
1
answer
287
views
Extension of first order deformations of a line bundle
Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
7
votes
2
answers
381
views
Deformation of (locally) ringed spaces and of their abelian categories of modules
I am interested in the general theory of deformations locally ringed spaces in the same language of the deformation theory of schemes/varieties that is already widely available. I am aware for example ...
- 1,482
1
vote
0
answers
155
views
Homogeneous deformation of isolated singularities
Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...
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 $m$...
1
vote
0
answers
149
views
Obstruction to deforming vector bundles
Let $X$ be a complex algebraic variety and let $D$ denote any $\mathbb C[[h]]$-deformation of $\mathcal O_X$. Suppose that $D$ is trivial. Then it is well-known that obstructions to deforming any $X$-...
3
votes
1
answer
160
views
Examples of jumping base locus of complete linear systems
I am looking for examples of invertible sheaves in smooth, projective families such that the associated base locus (i.e., the intersection of all the effective divisors in the complete linear system) ...
- 1,981
1
vote
0
answers
174
views
Comparison of logarithmic deformations and normal deformations
(I'm trying to learn logarithmic geometry, and I'm extremely confused about something very basic in log deformation theory. It's very likely my question is nonsense, but I don't get it.)
Let's pick ...
- 49
1
vote
0
answers
210
views
Ex 1.1c Hartshorne Deformation Theory: Is this family flat?
This comes from my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ...
- 161
6
votes
1
answer
491
views
Flatness of schemes
I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes ...
- 637
2
votes
0
answers
123
views
Consequences of smoothability
I have seen that there is a lot of work on studying the smoothable component of the Hilbert scheme of points $\textit{Hilb}^n(X)$ of some variety $X$. The main results are that if $\dim X \leq 2$ then ...
4
votes
1
answer
377
views
Deformation theoretic argument on dimension counting of naive Hurwitz scheme
I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves
and have a question about a suggested modification of an dimension
countinging argument applying ...
- 6,018
2
votes
1
answer
261
views
Smooth, non-isotrivial fibration with vanishing Kodaira-Spencer map at a point
This question arose by reading the paper [1], in particular, the remark at p. 737:
As an example, consider a non-isotrivial smooth projective morphism $f \colon X \to Y$ from a smooth projective ...
- 66.3k
3
votes
0
answers
222
views
Deformations of genus g curves to 'non-reduced rational curve'
We work over the complex numbers. Fix a genus $g$. Does there exist a connected reduced base $ B $ and a flat projective family $ \pi : X \rightarrow B $ satisfying the following two conditions?
its ...
- 936