The deformation-theory tag has no wiki summary.

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### Upper bound on the dimension of the Hilbert scheme of space cuves

Denote by $H_{P,Q}$ the flag Hilbert scheme parametrizing a pair $(C,X)$ such that $X$ is a degree $d$ surface in $\mathbb{P}^3$ with Hilbert polynomial $Q$ and $C \subset X$ is a curve with Hilbert ...

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**3**answers

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### Why are derived categories natural places to do deformation theory?

It seems to me that a lot of people do deformation theory (of schemes, sheaves, maps etc) in derived category (of an appropriate abelian category). For example, the cotangent complex of a morphism ...

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228 views

### 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 ...

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309 views

### Grothendieck-Messing theory

Hello,
I would like to work out some examples of deformation of isogenies via Grothendieck-Messing theory. Let's take an easy example: Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$ and ...

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**1**answer

178 views

### linear system of non-reduced divisor and associated reduced divisors

Let $X$ be a smooth degree $d$ $(d \ge 5)$ surface in $\mathbb{P}^3$. Let $D$ be an effective Cartier divisor (hence locally of complete intersection) on $X$ and $D_{red}$ the associated reduced ...

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**1**answer

244 views

### on a Deformation long exact sequence of moduli space of stable maps

I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence
\begin{align}
0 & \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, ...

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**1**answer

218 views

### The proof of unobstructedness of deformations for curves

I am reading Illusie's lecture notes "topics in algebraic geometry", and I have difficulty in following his proof of unobstructedness of deformation of curves. Here is the statement of the ...

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234 views

### Hilbert function of a Hilbert scheme

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the ...

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**4**answers

518 views

### Higgs mechanism from a deformation quantization point of view

Is it possible to describe the Higgs mechanism from a deformation quantization point of view? How would one do it? Are there aspects of the Higgs mechanism and Higgs particle which one may see clearer ...

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**1**answer

347 views

### base-point free linear system

Let $C$ be a reduced curve in a smooth degree $d$ ($d \ge 5$) surface in $\mathbb{P}^3$. Suppose $C=C_1 \cup ... \cup C_r$ with $C_i$ irreducible and $C_i^2<0$ for all $i$. Then is the linear ...

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233 views

### 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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198 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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**1**answer

262 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

118 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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487 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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166 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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137 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

169 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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239 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

289 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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**2**answers

265 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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286 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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**2**answers

277 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

316 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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303 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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### 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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### 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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282 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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315 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

337 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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**2**answers

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### 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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**1**answer

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### 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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### 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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342 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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### 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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### 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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### 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

565 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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### Algebraic vs. topological study of deformations — deformation stacks vs. topologized deformation spaces

In my limited understanding, one can always find a category that captures the data of a deformation problem. But for a given deformation problem with a topologized deformation space, is there any ...

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141 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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### 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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391 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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286 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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### 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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545 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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### 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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### 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 ...