# Tagged Questions

The deformation-theory tag has no usage guidance.

**2**

votes

**2**answers

295 views

### General degree $d$ surface in $\mathbb{P}^3$

Let $H_{d_1,g_1}, H_{d_2,g_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d_1, d_2$ and genus $g_1, g_2$. Denote by $H:=H_{d_1,g_1}\times H_{d_2,g_2}$
where an element in $H$ is ...

**3**

votes

**2**answers

336 views

### Infinitesimal rigidity vs. local rigidity

I am thinking about homomorphisms $\mathrm{Hom}(\Gamma,G)$, where $G$ is a Lie group and $\Gamma$ is a discrete, finitely generated subgroup.
This question talked about the difference of ...

**2**

votes

**2**answers

288 views

### Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups

Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak ...

**7**

votes

**1**answer

467 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 ...

**1**

vote

**2**answers

391 views

### Any irreducible component of the HIlbert scheme contains an irreducible element

Consider $Hilb_{d,g}$ the Hilbert scheme of curves in $\mathbb{P}^3$ of degree $d$ and genus $g$. Is it true that if $L$ is an irreducible component of $Hilb_{d,g}$
then there exists a curve $C \in L$ ...

**2**

votes

**1**answer

184 views

### What is the definition of “the $L_\infty$ part of a $G_\infty$ morphism”?

We know that in Tamarkin's proof of Kontsevich's formality theorem, he defined the $G_\infty$ structure on the Hochschild cochain complex $C^\cdot(A,A)$ and constructed a $G_\infty$ morphism from ...

**5**

votes

**1**answer

676 views

### Examples of nice reduced singularities on Hilbert schemes--Edited

In his "Murphy's Law" paper, Vakil showed that every "singularity type" (with a precise meaning) occurs on certain Hilbert schemes; for instance, the Hilbert scheme of nonsingular curves in projective ...

**0**

votes

**1**answer

393 views

### T^1 functors and Ext^1 in deformation theory

Why are the first-order deformations of a scheme $X$ over a field $k$ given by $Ext^1_{\mathcal O_X}(\Omega_X,\mathcal O_X)$, where here I mean the Ext group?
Furthermore, for an integral affine ...

**4**

votes

**1**answer

448 views

### Hodge numbers of a Calabi-Yau 3-fold via deformation theory

In their paper "Calabi-Yau 3-folds and Moduli of Abelian Surfaces I", Gross and Popescu calculate the hodge numbers of smooth CY 3-folds obtained in the following way (see Remark 4.11 of their paper): ...

**0**

votes

**1**answer

398 views

### Irreducible components of the Hilbert scheme

Let $P_1, P_2, Q$ denote the Hilbert scheme of a plane conic in $\mathbb{P}^3$, a quartic and a degree $d$ surface in $\mathbb{P}^3$. Then there is a natural inclusion map $i$ from Hilbert flag scheme ...

**19**

votes

**1**answer

1k 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 ...

**2**

votes

**2**answers

478 views

### Lattices in algebraic geometry

In Chriss and Ginzburg's Complex Geometry and Representation Theory, they define a lattice in the following setting: let $A$ be a ring (I guess I only care about the case that $A$ is a commutative ...

**2**

votes

**1**answer

237 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 ...

**5**

votes

**1**answer

468 views

### Has the cotangent complex been used in context other than morphism of schemes?

Here is what I know about the history of the cotangent complex:
Quillen did it over a point (i.e. for morphism of rings), Illusie did it in a topos (i.e. for sheaves of rings in a topos). And proved ...

**6**

votes

**1**answer

480 views

### what can be reached by flat degeneration of (globally) complete intersection?

Let $X\subset\mathbb{P}^n$ be a (globally) complete intersection, let $(X_t)_{t\in\mathbb{C}^1}$ be a flat family, with $X_1=X$. Which types of schemes can we get as $X_0$?
Or, conversely, which ...

**1**

vote

**2**answers

754 views

### On the algberaicity of the universal elliptic curve associated to a torsion free subgroup

So let $\Gamma\subseteq SL_2(\mathbf{Z})$ be a finite index subgroup (not necessarily a congruence subgroup). Recall that we have an action of $SL_2(\mathbf{R})$ on
...

**2**

votes

**0**answers

180 views

### Smooth curve in the Hilbert flag scheme

Let $d$ be an integer greater than $0$. Let $P_2$ be the Hilbert polynomial of a degree $d$ surface in $\mathbb{P}^3$. Recall, the Hilbert flag scheme $\mathrm{Hilb}_{P_1,P_2}$ parametrizes curves $C$ ...

**13**

votes

**1**answer

473 views

### Les deux théorèmes d'existence en théorie formelle des modules

In Exposé 195 of the Séminaire Bourbaki, Grothendieck states the following two theorems of non-flat descent.
Theorem 1. Let $\Lambda$ be a noetherian ring and $C$ the category of ...

**12**

votes

**1**answer

707 views

### Lie groups vs. algebraic groups and deformations

I am interested in deformations of (discrete subgroups of) Lie groups. But, as I understand it, deformation theory, as a theory, prefers to speak schemes.
At least the classical Lie groups can be ...

**1**

vote

**1**answer

142 views

### Linear equivalence and Hilbert function

Let $X \subset \mathbb{P}^3$ be a smooth degree $d$ surface containing two irreducible curves $C_1, C_2$ linearly equivalent to each other. If we assume that $X$ is general (among all degree $d$ ...

**1**

vote

**0**answers

210 views

### Do deformations of isolated hypersurface singularity naturally induce deformations of their divisors?

Let $0 \in V =(f=0) \subset \mathbb{C}^{n+1}$ be an affine variety with an isolated hypersurface singularity at the origin for $n \ge 3$.
Let $0 \in D=(x=f=0) \subset V$ be a divisor with only ...

**3**

votes

**2**answers

513 views

### Degeneration of projective curves

Given a projective curve $C$, is it possible that $C$ can degenerate into union of lines i.e., does there exist a family of curves $\pi:\mathcal{C} \to B$ such that $\pi^{-1}(0)=C$ and there exists $a ...

**2**

votes

**0**answers

96 views

### Dual Honda systems

Hello,
There is an equivalence of categories between p-divisible groups over the ring of Witt vectors $W(k)$ and the category of "Honda systems", that is couples $(M,L)$ formed by a Dieudonné module ...

**2**

votes

**1**answer

292 views

### Extending smooth irreducible representations

Hi,
Let $G_1, G_2$ be topological groups with $G_1 \subset G_2$ is closed. Let $\rho:G_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea ...

**3**

votes

**0**answers

278 views

### lifting abelian varieties

Hello,
Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$. This abelian variety is endowed with a principal polarization and an action by $\mathcal{O}_E$, the ring of integers of an ...

**17**

votes

**2**answers

896 views

### Massey Products vs. $A_\infty$-Structures

Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More ...

**4**

votes

**1**answer

504 views

### Deformation space of non-ordinary abelian varieties

It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...

**1**

vote

**2**answers

783 views

### When do infinitesimal deformations lift to global deformations?

Hi,
I understand the notion of Infinitesimal and global deformations and the fact that global deformations lead to certain infinitesimal deformations. But I could not find any criterion or idea to ...

**4**

votes

**0**answers

381 views

### Mazur's relative deformation functors

In his paper "Deformation Theory of Galois Representations" in the book "Modular Forms of Fermat Last Theorem", Mazur considers more general deformation functors than in his earlier
paper in "Galois ...

**4**

votes

**2**answers

345 views

### $H^1$ of the pull back of the tangent bundle.

If $C$ is a smooth elliptic curve and $f: C \to \mathbb P^n$, then
$H^1(C,f^*T_{\mathbb P^n}) = 0.$ How do I prove this? The implication is that map
from $C$ to $\mathbb P^n$ is unobstructed.

**18**

votes

**4**answers

1k views

### A matrix algebra has no deformations?

I have often heard the slogan that "a matrix algebra has no deformations," and I am trying to understand precisely what that means. While I would be happy with more general statements about ...

**5**

votes

**1**answer

531 views

### Why are people interested in Cohen-Macaulay of codimension 2?

In deformation theory, Cohen-Macaulay in codimension 2 is the first to be considered in higher order deformation. Does Cohen-Macaulay in codim. 2 have some good property to work with? Does it somehow ...

**3**

votes

**5**answers

305 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 ...

**16**

votes

**1**answer

1k views

### Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 3

Part 3 of this series of questions. In the meantime, I realized that there is some very simple question that was left open in Accumulation of algebraic subvarieties: Near one subvariety there are many ...

**5**

votes

**2**answers

705 views

### About Kodaira's book on deformations

I happened to read the following sentence in the blog by the physicist Jacques Distler:
"What makes Kodaira’s Complex Manifolds and Deformation of Complex Structures such a delight to read is that ...

**3**

votes

**1**answer

461 views

### Algebra Deformations and Maurer-Cartan elements

Hello to all,
If $(A,\mu)$ is an algebra, it is very well known that set of deformations mod equivalence is isomorphic to the of Maurer-Cartan set of the DG Lie algebra of the hochschild cocomplex ...

**2**

votes

**2**answers

574 views

### Tamarkin-Tsygan Formalism

Hello to all.
There is a well-known formalism in deformation-quantization which puts the algebraic structure of polyvector fields in a noncommutative setting. Tamarkin-Tsygan define a (pre)calculus ...

**5**

votes

**1**answer

733 views

### A simple proof of the Weyl algebra's rigidity.

I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found in ...

**1**

vote

**1**answer

565 views

### Galois Cohomology maps

Let $\bar{\rho}$ be a residual ordinary and locally split galois representation associated to a weight $k$ level $1$ form (more specifically a mod $p$ companion form). In the sense of deformation ...

**14**

votes

**2**answers

2k views

### obstruction theories in algebraic geometry

I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction ...

**10**

votes

**1**answer

778 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**

votes

**2**answers

2k views

### Accumulation of algebraic subvarieties: Near one subvariety there are many others (?)

Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the ...

**7**

votes

**3**answers

902 views

### In the dictionary between Poisson and Quantum, what corresponds to Coisotropic?

I work entirely over a field of characteristic $0$, in case it matters.
Recall that a Poisson algebra is a commutative algebra $A$ with a bracket $\lbrace,\rbrace : A^{\wedge 2} \to A$ which is (1) a ...

**16**

votes

**2**answers

698 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?

**6**

votes

**2**answers

847 views

### Relationship between Hilbert schemes and deformation spaces

Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...

**9**

votes

**1**answer

383 views

### Reverse Engineering to find deformation problem (from cohomology groups)?

One of my favorite explanation of the cohomology groups of low degree is that they arise as the automorphism group, tangent space and obstruction space (or where the obstruction lives) of a certain ...

**10**

votes

**0**answers

446 views

### Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...

**10**

votes

**1**answer

920 views

### Kontsevich's formality theorem from an explicit homotopy

Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators ...

**4**

votes

**1**answer

341 views

### Extension of a first order deformation of a sheaf

Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.
Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.
Assume all ...

**8**

votes

**1**answer

951 views

### About the Serre-Tate theorem

It is somehow a general principle that the (infinitesimal) local behavior of a representable moduli functor $X$ at some point $x$ is closely related to the deformation problem of the structure ...