# Tagged Questions

**4**

votes

**2**answers

350 views

### Are linear algebraic groups rigid?

The underlying variety of a linear elgebraic group (say, over an algebraically closed field) is affine, so doesn't have nontrivial (infinitesimal) deformations. I'm curious to know whether it's ...

**3**

votes

**0**answers

110 views

### On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...

**0**

votes

**1**answer

181 views

### Obstruction and 1st order infinitesimal deformations of Generalized Elliptic Curves (Deligne-Rapoport)

We consider the deformation theory of a generalized elliptic curve $(C_0,+)$ over a field $k$. Let $D$ be the deformation functor.
And now we only consider the case that $C_0$ is irreducible as in ...

**2**

votes

**1**answer

94 views

### Generic vs General property of reducedness in a family of projective schemes

Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a generic point on $B$. Denote by $\mathcal{X}_K$, the pull-back of $\mathcal{X}$. This ...

**1**

vote

**0**answers

77 views

### Obstruction to Gorenstein Liaisons of space curves

Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and ...

**4**

votes

**1**answer

174 views

### Surjectivity of certain cohomology groups on hypersurfaces of high degree

I had been reading an article by Spencer Bloch. There is a remark in this text which states the surjectivity of a particular map between cohomology groups without explaining further. I had been trying ...

**5**

votes

**1**answer

478 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 ...

**0**

votes

**1**answer

130 views

### Is being fano preserved under flat base change [closed]

Let $K$ be a field of characteristic zero and $X$ be a projective variety over $\mbox{Spec} K$. Denote by $\bar{K}$ the algebraic closure of $K$. Let $\bar{X}$ be the fiber product $X \times_K ...

**4**

votes

**1**answer

150 views

### Simultaneous resolution of singularities in special cases of flat families of projective varieties

Let $\pi:\mathcal{X} \to B$ be a flat family of projective varieties. Assume that $B$ is irreducible. Suppose that $\mathcal{X}$ is smooth except for a closed subscheme, say $Y$ which is isomorphic to ...

**3**

votes

**1**answer

113 views

### Deforming curves to other curves over the field of rational numbers

Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.
If $k$ is an algebraically closed field of characteristic zero, there exists a connected ...

**0**

votes

**1**answer

64 views

### Two questions about line bundles over Kuranishi families

i'm studying the article "Variétés Kahleriennes dont la première classe de chern est nulle" by Arnaud Beauville and i have a couple of questions i would like to ask you, hoping they are not too ...

**3**

votes

**1**answer

276 views

### Cohomology and proper base change

Let $\pi:\mathcal{X} \to B$ be a flat, projective surjective morphism over $\mathbb{C}$. Assume that $B$ is a smooth quasi-projective curve. Let $\mathcal{F}$ be a coherent sheaf on $\mathcal{X}$, ...

**1**

vote

**1**answer

149 views

### On infinitesimal neighbourhood of a point in a projective scheme

Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$ and $X \subset Y$. Let $x \in X$ be a closed point. Assume that for any positive integer $n$ and any morphism from $\mathrm{Spec} ...

**4**

votes

**0**answers

243 views

### Spectral sequences and Koszul complexes in Deformation Theory

I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper):
Let $A'$ be a noetherian local ring with ...

**1**

vote

**1**answer

235 views

### Are deformations of quotients of local rings embedded?

In Hartshorne's book "Deformation Theory" one can find a statement (inside the proof of Theorem 10.1) that every deformation $X' \to Spec(C)$ of an affine scheme $$X = ...

**1**

vote

**1**answer

144 views

### Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$.
Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...

**3**

votes

**0**answers

113 views

### Cycle class map in non-smooth family of projective varieties

Let $\pi:\mathcal{X} \to T$ be a family of smooth projective complex varieties. Assume $T$ is quasi-projective, reduced, irreducible but not smooth and of positive dimension. Let $\mathcal{Z}$ be a ...

**6**

votes

**0**answers

243 views

### A question on infinitesimal deformation (related to intersection theory)

Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...

**2**

votes

**1**answer

118 views

### Locally trivial deformations of surfaces with quotient singularities

Let us consider the surface $\mathbb{A}^{2}/\mu_{6}$ where the action is given by
$$
\begin{array}{ccc}
\mu_{6}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,x_{1},x_{2}) ...

**1**

vote

**0**answers

87 views

### Deformation of complete intersection curves

Let $\mathcal{X} \to B$ be a family of smooth surfaces in $\mathbb{P}^3$. Let $\mathcal{C} \to B$ be a family of curves in $\mathbb{P}^3$. Assume that for all $b \in B$, the fiber $\mathcal{C}_b$ is ...

**2**

votes

**0**answers

151 views

### Stacks and Maurer-Cartan elements

One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$.
For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...

**3**

votes

**1**answer

160 views

### Deformations of quotient singularities

Let $Y$ be an affine scheme over a field of characteristic zero. Suppose we have a group $G$ acting on $Y$ and that the subset of $Y$ of points with non-trivial stabilizer is in codimension greater or ...

**8**

votes

**0**answers

197 views

### Are deformations of a scheme some kind of a “derived gerbe” under the cotangent complex?

(This is probably a very naive question. My understanding of the cotangent complex is quite vague.)
Let me first recall the picture for deformations of a smooth morphism:
If $f:X_0\to S_0$ is a ...

**8**

votes

**1**answer

359 views

### Why can you deform singularities in two dimensions but not in higher dimensions?

I've been trying to read this paper to understand deformations of surface quotient singularities. I'm particularly interested in when one can deform certain cyclic quotient singularities into other ...

**4**

votes

**1**answer

132 views

### Characterizing the rigidity of morphisms of smooth varieties

Let $X$ and $Y$ be smooth algebraic varieties over a field $k$ of characteristic $0$. For varieties we know that $X/k$ is rigid if and only if $H^{1}(X,T_{X})=0$. But $H^{1}(X,T_{X})$ also ...

**2**

votes

**1**answer

130 views

### Deformation of transversal intersection

Fix a positive integer $n \ge 2$. Let $\pi:\mathcal{X} \to B$ be a family (flat, projective and surjective morphism) of projective subschemes of $\mathbb{P}^n$.
Assume $B$ is reduced, irreducible.
...

**5**

votes

**1**answer

210 views

### A question on the morphism between Hilbert schemes

Let $L_1,L_2$ be two irreducible component of two different Hilbert schemes parametrizing closed subscheme in $\mathbb{P}^n$ and $\mathbb{P}^{n-1}$, respectively. Denote by $\pi_1: \mathcal{X}_1 \to ...

**7**

votes

**1**answer

160 views

### Projective embedding in families of curves

Let $\pi:\mathcal{X} \to B$ be a family (flat, projective, surjective morphism) of projective curves (not necessarily reduced) where $B$ is smooth, irreducible. Suppose that for some closed point $b_0 ...

**2**

votes

**0**answers

102 views

### Projective deformations of a projective hyperkahler manifold

Let $X$ be a projective hyperkahler manifold, $f:\mathcal{X}\rightarrow \mathcal{M}$ its Kuranishi family. We define $\mathcal{P}:=\{m\in \mathcal{M} $such that $\mathcal{X}_m$ is projective$\}$, ...

**4**

votes

**1**answer

192 views

### Relating deformations of a scheme to deformations of its singular locus

Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...

**4**

votes

**1**answer

129 views

### Projective embedding of curves which preserves the degree

Let $C$ be a projective curve (not necessarily reduced or irreducible).
Let $f:C \hookrightarrow \mathbb{P}^n$ be a closed immersion. Suppose that the degree of $f(C)$ is equal to $e$. How many ways ...

**3**

votes

**1**answer

153 views

### Generalization of Hilbert/Quot schemes

For some positive integer $n$, recall that the Quot scheme $Quot(\mathcal{O}_{\mathbb{P}^n})$ parametrizes ideal sheaves of subschemes in $\mathbb{P}^n$. As far as I understand (from a previous post) ...

**1**

vote

**1**answer

215 views

### Making a family of schemes non-reduced

Let $f:X \to Y$ be a flat, family of smooth projective varieties. Assume futher that $Y$ is smooth. Suppose there exists a scheme $Y'$ such that the associated reduced scheme $Y'_{\mathrm{red}} \cong ...

**4**

votes

**0**answers

208 views

### Infinitesimal deformation and contractibility of algebraic curves

Let $X$ be a smooth projective surface and $X'$ be an infinitesimal deformation of $X$. Denote by $f: X \to X'$ the natural closed immersion. Let $C' \subset X'$ be a curve such that $f^{-1}(C')$ is ...

**2**

votes

**1**answer

154 views

### Are singular rational curves on K3 surfaces rigid?

Let $S$ be a K3 surface over the complex numbers $\mathbb{C}$. If $C\subset S$ is a smooth rational curve, the normal bundle $N_{C/S}$ is isomorphic to $\mathbb{O}(-2)$ and thus $C$ is rigid. What ...

**1**

vote

**1**answer

253 views

### A question on nested Hilbert scheme

Let $H_1, H_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n_1}, \mathbb{P}^{n_2}$ with Hilbert polynomials $P_1, P_2$, respectively. Given a pair $(Z_1, Z_2)$ of subschemes in ...

**2**

votes

**1**answer

212 views

### Is Mazur's deformation ring R integral?

Consider the absolutely irreducible Galois representation
$\overline{\rho} \colon G_{\Bbb Q} \to {\mathrm{GL}}_2({\Bbb F}_p)$. We apply the Mazur's deformation theory on the lift ${\rho} \colon ...

**3**

votes

**0**answers

254 views

### Deformation of a family of curves in a surface

Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for ...

**2**

votes

**1**answer

243 views

### Existence of rational section to a flat projective morphism

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of dimension greater than $1$. Suppose there exists a flat projective morphism $f:X \to \mathbb{P}^n$ for some $n \ge 1$. ...

**1**

vote

**1**answer

104 views

### Rational equivalence and infinitesimal deformation of curves

Let $C_1$ and $C_2$ be two rationally equivalent curves in $\mathbb{P}^3$. Is it true that the dimension of $H^0(\mathcal{N}_{C_1|\mathbb{P}^3})$ equal to that of ...

**2**

votes

**0**answers

64 views

### Changing the Hilbert scheme of curves by adding the hyperplane section

Let $X$ be a smooth degree $d$ ($d>4$) surface in $\mathbb{P}^3$.
Let $C$ be a divisor on $X$. Let $D$ be a general element of the linear series $|C+H_X|$ where $H_X$ is the hyperplane section of ...

**0**

votes

**0**answers

73 views

### Surjectivity on tangent space of Hilbert schemes imply dominance?

Let $P_1, P_2$ be two Hilbert polynomials of certain subschemes in $\mathbb{P}^n$ and $Hilb_{P_1,P_2}$ be the flag Hilbert scheme parametrizing all pairs $(Z_1,Z_2)$ such that $Z_1 \subset Z_2$ and ...

**2**

votes

**0**answers

237 views

### On infinitesimal deformation of projective varieties

Let $X$ be a smooth complex projective variety. Suppose $X \hookrightarrow \mathbb{P}^n$. Let $Z$ be a closed (reduced) subscheme of $X$.
Let $X'$ be an infinitesimal deformation of $X$ corresponding ...

**1**

vote

**1**answer

126 views

### Isomorphism of homology groups under deformation

Let $\pi:\mathcal{X} \to U$ be a family of hypersurfaces (not necessarily smooth) in $\mathbb{P}^n$ for some $n \ge 3$. Assume that $U$ is simply connected (under analytic topology). For any pair $u,v ...

**2**

votes

**0**answers

280 views

### Deformation over small disk and deformation over complex disk

Let $D=\mathop{Spec}(\mathbb C[[t]])$ be the algebraic small disk and let $\Delta= \{z\in \mathbb C: |z|<1\} $. Let $X_0$ be an algebraic surface over $\mathop{Spec}\mathbb C$
Suppose now that I ...

**4**

votes

**1**answer

314 views

### Why should we study deformations of perfect complexes

What are the advantanges of studying deformation of perfect complexes over the classical theory of deformation of coherent sheaves? Any references which elaborates on the applications on deformation ...

**1**

vote

**0**answers

95 views

### Functorial property of universal family

Let $P_1, P_2$ and $P_3$ be Hilbert polynomials of projective schemes. Suppose that the flag Hilbert scheme $Hilb_{P_1,P_2}$ is non-empty. Assume further that there exists a closed immersion of the ...

**4**

votes

**1**answer

424 views

### Is projective morphism with projective fiber flat?

Let $X, Y$ be quasi-projective Noetherian schemes and $f:X \to Y$ be a projective surjective morphism. Assume that every fiber of $f$ is isomorphic to a projective space $\mathbb{P}^n$ for a fixed ...

**1**

vote

**0**answers

80 views

### cotangent complex for finite flat morphism and different ideal

Let a ring $A$, $F=A[X_{1},\dots X_{n}]$ and $B:= F/J$.
We suppose that we have a finite flat lci morphism $f:Spec(B)\rightarrow Spec(A)$.
To mesure the singularities of this map, Gabber-Ramero ...

**1**

vote

**0**answers

151 views

### Contractibility of union of smooth rational curves in famillies

Let $\pi:\mathcal{X} \to S$ be a family of smooth surfaces containing a subfamily $\mathcal{Y} \subset \mathcal{X}$ proper, flat over $S$ parametrizing contractible curves in $\mathcal{X}$ satisfying ...