The deformation-theory tag has no usage guidance.

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### Hilbert scheme of curves in a degree $d$ surface in $\mathbb{P}^3$

Fix a Hilbert polynomial $P$ of a non-plane curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of ...

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230 views

### Deformation of modules over noncommutaitve rings

Let $M$ be a finitely generated module over a commutative ring $R$. The first order deformation of module $M$ is parametrized by $Ext^{1}(M,M)$ and the obstruction is parametrized by $Ext^{2}(M,M)$. ...

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151 views

### Deformation of Noether's first theorem

Noether's first variational theorem establishes a correspondence between symmetries and invariants. I would like to know what has been written on the following question: How do the invariants deform ...

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votes

**1**answer

311 views

### Deformation of space curves to union of lines

Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ ...

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**1**answer

125 views

### Rational equivalence and Hilbert flag scheme

Given a smooth surface $X \subset \mathbb{P}^3$ if we have two curves $C_1, C_2$ that are rationally equivalent is it true that both $(C_1,X)$ and $(C_2,X)$ will be in the same irreducible component ...

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847 views

### How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra?

First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that ...

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172 views

### dimension of spaces of rational curves in a variety!

Do you know calculate the dimension of the space of rational curves of degree
m, through d given points, contained in some projective variety?

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150 views

### Maximum number of generators of a curve in $\mathbb{P}^3$

Let $H_{d,g}$ denote the Hilbert scheme of curves of degree $d$ and genus $g$ locally of complete intersection in $\mathbb{P}^3$. Given a curve $C \in H_{d,g}$, denote by $S(C)$ a minimal set of ...

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**1**answer

241 views

### When is the natural projection of the HIlbert flag scheme a flat morphism

Let ${Hilb_{P,Q}}_{red}$ be the reduced scheme associated to the Hilbert flag scheme parametrizing all pairs $(C,X)$ with
$C \subset X \subset \mathbb{P}^3$, where $C$ is a curve and $X$ a degree $d$ ...

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305 views

### triangle group representation

Let $G_{\alpha,\beta,\gamma}$ be a hyperbolic triangle group. Then $G$ has a following presentation.
$G=\langle a, b; a^{\alpha}, b^{\beta}, c^{\gamma}, abc\rangle, \alpha, \beta, \gamma \geq 2$.
...

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**1**answer

307 views

### T^i functors are isomorphic for analytically isomorphic isolated singular points

I've been having trouble proving the following:
Let $B$ and $B'$ be local rings, essentially of finite type over $k$, having isolated singularities at the closed points. Suppose that they are ...

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298 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 ...

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342 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 ...

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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 ...

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**1**answer

490 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 ...

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410 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 $C \in L$ ...

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**1**answer

186 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 ...

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**1**answer

684 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 ...

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**1**answer

402 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 ...

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**1**answer

461 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): ...

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**1**answer

409 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 ...

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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 ...

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483 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 ...

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239 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 ...

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**1**answer

475 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 ...

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506 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 ...

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764 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
...

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186 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 $C$ ...

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479 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 ...

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**1**answer

717 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 ...

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144 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 $d$ ...

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212 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 ...

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528 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 ...

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### 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 ...

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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 ...

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282 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 ...

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950 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 ...

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**1**answer

513 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 ...

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798 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 ...

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### 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 ...

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**2**answers

346 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.

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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 ...

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**1**answer

543 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 ...

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**5**answers

308 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 ...

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**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 ...

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### 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 ...

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481 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 ...

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584 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 ...

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874 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 ...

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568 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 ...