# Tagged Questions

**2**

votes

**1**answer

132 views

### deformations of vector bundles on curves

Let $X$ be a smooth algebraic curve. Suppose I have a flat family $V_y\to X$ of vector bundles on $X$ over an affine scheme $S$. Let $p=Spec(k)$ be one geometric point of $S$. If the determinant of ...

**6**

votes

**1**answer

502 views

### Kodaira-Spencer theory of deformation done right

I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of ...

**2**

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**0**answers

108 views

### Family with a fixed special fiber over finite fields

Let $X$ be a smooth projective variety over a finite field $\mathbb{F}_p$. What are the conditions for the existence of a projective variety $X'$ over $\mathbb{Q}_p$ such that $X$ is a special fiber ...

**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 ...

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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 ...

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**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 ...

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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 ...

**6**

votes

**2**answers

458 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 ...

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**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 ...

**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(Σ, p_1, . . . , p_n, f)\to Aut(Σ, ...

**0**

votes

**1**answer

375 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

**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**

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**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 ...

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**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 $d$ ...

**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 ...

**6**

votes

**1**answer

344 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 ...

**2**

votes

**1**answer

219 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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**0**answers

157 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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424 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 ...

**12**

votes

**5**answers

2k views

### Deformations of semisimple Lie algebras

In the questions Is "semisimple" a dense condition among Lie algebras? and What is the Zariski closure of the space of semisimple Lie algebras?, something equivalent to the following is ...

**4**

votes

**2**answers

636 views

### Versality in deformation theory vs. versality in moduli spaces

As I mentioned before, I'm a novice at deformation theory. I was wondering if the definition of versality in deformation theory is related to the versality in moduli spaces:
Deformation theory
...

**18**

votes

**5**answers

2k views

### Is a 'generic' variety nonsingular? Or singular?

I'd like to know whether there's some coherent meaning of 'generic' for which one can say that a 'generic' variety over an algebraically closed field $K$, say, is nonsingular or singular. We could ...

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**0**answers

557 views

### “a theory of generalized Donaldson-Thomas invariants” by Joyce & Song

Is anyone else working through this paper : http://arxiv.org/abs/0810.5645 ? I am trying to verifying example 6.2 (m=2 for simplicity) using only the definitions, namely:
J^{2\alpha}(\tau) = -1/4
...