The deformation-theory tag has no wiki summary.

**4**

votes

**1**answer

245 views

### Local model of virtual fundamental cycle

The following baby version of virtual fundamental cycle is well known:
Let $M\subset V$ be the zero locus of a section $s$ of a vector bandle $E \to V$, in general $s$ is not transversal to the zero ...

**2**

votes

**1**answer

143 views

### does there exist a family of objects over the tangent space to the base space of a family of objects?

Suppose we have a family of objects $\Xi \to S$ over a base smooth projective scheme $S$. Take a closed point $p\in S$ and consider the tangent space to $S$ at $p$. Can one construct an "induced ...

**1**

vote

**1**answer

110 views

### Zero Sums in a $q$-Deformation Remain Zero for $q=1$

Looking at previous question, I begun to think and came upon the following more solid question: Let $SU_q(N)$ be the usual quantized coordinated algebra of $SU(N)$. Now consider, for some fixed value ...

**1**

vote

**2**answers

211 views

### Classification of first order deformations of n-pointed non-singular variety

Why is the set of first order deformations equal to $H^1(X,T_{X}(-p_1 -p_2 ... -p_k))$? And what is the motivation for studying this? (I know the proof for unpointed curves, which can be found in a ...

**3**

votes

**1**answer

341 views

### Formal criterion of flatness

Let $k$ be a field, $S$ and $R$ be local $k$-algebras with residue field $k$ and $\phi:S\to R$ be a local homomorphism. Then $\phi$ induces (obviously) a natural transformation of "functors of points" ...

**0**

votes

**1**answer

110 views

### Deformations and Dimensions: $q$-Deform Finite to Infinite?

Let $A$ be a (complex) (finitely generated) algebra with some type of $q$-deformation $A_q$, where $A_1 = A$. Moreover, let $V_q$ be a vector subspace of $A_q$, such that $V_1$ is a ...

**1**

vote

**1**answer

188 views

### Singular locus of a Hilbert scheme

Consider the Hilbert scheme $H$ of conics in $\mathbb{P}^3$. It is easy to see that there exists a closed subscheme $H'$ of $H$ parametrizing $2$ lines intersecting at a point. This can be seen as the ...

**1**

vote

**1**answer

184 views

### glueing flat families of objects over a blow-up

Hi Everybody,
I stumbled upon the following question for families of vector bundles (over a curve) but I guess it could be interesting to answer in general.
Suppose I have $B$ the blow-up of a ...

**4**

votes

**3**answers

313 views

### When is a smooth projective variety a fibration

Let $X$ be a smooth projective variety. Is there a criterion (apart from the definition) for the existence of a projective curve $C$ and a proper surjective morphism $\pi:X \to C$?

**7**

votes

**2**answers

434 views

### Algebraic definition of the Kuranishi map

Let $X$ be a smooth projective algebraic variety over an algebraically closed field $k$. If $k=\mathbb{C}$, we know by work of Kuranishi that the base of the versal deformation of $X$ is the germ at ...

**11**

votes

**1**answer

946 views

### Deformations of the punctured affine plane

Let $k$ be some field, algebraically closed and of characteristic $0$, if you like.
Let $U= \mathbb{A}^2_k \setminus \{ (0,0) \}$ be the punctured affine plane over $k$. Write $U$ as the union of ...

**6**

votes

**1**answer

184 views

### Obstruction sheaf is a vector bundle when the moduli space is non-singular?

I am working on some basic of Gromov-Witten theory and stuck in understanding obstruction bundle. Recall that a perfect obstruction theory on a scheme or stack $M$ due to Behrend and Fantechi is a ...

**1**

vote

**2**answers

241 views

### Infinitesimal deformations and moving cycles

The wonderful responses to an earlier question Self-intersection and the normal bundle motivated me to ask the following question:
Let $Y \subset X$ be a subvariety of a variety $X$. Infinitesimal ...

**1**

vote

**1**answer

97 views

### Upper bound on the dimension of linear series on a smooth hypersurface

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $d \ge 5$. Let $C \subset X$ be a reduced curve such that $C$ is not a complete intersection curve in $\mathbb{P}^3$. Is it true that ...

**0**

votes

**1**answer

93 views

### orthotropic materials solution of boundary value problems

What are the methods or approaches for the analytical solutions of boundary value problems in the theory of elasticity for orthotropic materials?

**2**

votes

**1**answer

194 views

### Techniques for showing that a curve is not smoothable

There are a number of techniques in algebraic geometry that can be used to show that a given reducible (often genus-zero) curve $C$ in a smooth variety $X$ becomes smooth and irreducible after a ...

**0**

votes

**0**answers

173 views

### tangent bundle of the toric variety of the wonderful compactification.

Let G be a adjoint group over $k$,algebraically closed of caracteristic zero.
Let $\overline{G}$ be its wonderful compactification.
I denote by $\overline{T}$ the closure of the torus $T$ in ...

**6**

votes

**2**answers

462 views

### Tangent space of the stack $\overline{\mathcal{M}}_{g,n}(X,\beta)$.

Let $\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the moduli stack of stable maps $f$ from genus $g$, $n$-marked curve $C$ to a variety $X$ i nth curve class $\beta \in H^2(X,\mathbb{Z})$. I would like ...

**0**

votes

**0**answers

145 views

### Normal sheaf of non-reduced locally complete intersection space curves

Let $C$ be a non-reduced locally complete intersection curve on a smooth degree $d$ surface in $\mathbb{P}^3$ (for example a non-reduced Cartier divisor). For simplicity we can assume that ...

**8**

votes

**1**answer

408 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 ...

**1**

vote

**1**answer

465 views

### Can any local complete intersection subvariety be an intersection of smooth hypersurfaces

Let $Z$ be a local complete intersection subscheme of dimension $m$ in $\mathbb{P}^{2m+1}$. Let $P$ be the Hilbert polynomial of $Z$. Denote by $Hilb_P$ the Hilbert scheme of local complete ...

**6**

votes

**0**answers

136 views

### Def-Obs theory of sheaves with fixed determinant on CY3.

Let $\mathcal{E}$ be a stable sheaf on a smooth complex projective threefold $X$ and $Ext^k_0(\mathcal{E},\mathcal{E})$ be the traceless Ext groups, defined by the kernel of the trace map
$$
...

**2**

votes

**0**answers

128 views

### which deformation of a matrix lead to flat deformations of determinantal varieties (fitting ideals)?

Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, ...

**3**

votes

**1**answer

243 views

### 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 ...

**18**

votes

**3**answers

2k views

### 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 ...

**5**

votes

**0**answers

240 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 ...

**2**

votes

**0**answers

339 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 ...

**0**

votes

**1**answer

181 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 ...

**6**

votes

**1**answer

263 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(ÎŁ, p_1, . . . , p_n, f)\to Aut(ÎŁ, ...

**3**

votes

**1**answer

227 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 ...

**3**

votes

**0**answers

240 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 ...

**2**

votes

**4**answers

529 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 ...

**0**

votes

**1**answer

376 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 ...

**2**

votes

**0**answers

262 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 ...

**2**

votes

**0**answers

206 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 $Ext^{2}(M,M)$. ...

**2**

votes

**1**answer

281 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 $C \subset \mathbb{P}^3$ ...

**0**

votes

**1**answer

121 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 ...

**3**

votes

**2**answers

535 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 ...

**1**

vote

**0**answers

167 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?

**0**

votes

**0**answers

138 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 ...

**1**

vote

**1**answer

185 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 $d$ ...

**3**

votes

**0**answers

255 views

### triangle group representation

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

**2**

votes

**1**answer

292 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 ...

**2**

votes

**2**answers

276 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

295 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

279 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 ...

**6**

votes

**1**answer

346 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

321 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

179 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

632 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 ...