The deformation-theory tag has no wiki summary.

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### 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 ...

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### 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.
...

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### 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 ...

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172 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 ...

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107 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$\}$, ...

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### 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 $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...

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### 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 ...

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162 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) ...

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220 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 ...

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210 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 ...

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### 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 ...

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262 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 ...

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220 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 ...

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259 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 ...

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257 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 $n \ge 1$. ...

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### 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 ...

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### Higher commutators in E_n algebras and the Maurer--Cartan equation

Let $A$ be an associative algebra in $dgVect_k$. Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra. The ...

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### 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 ...

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181 views

### Deformations of a complex trivial up to quasi-isomorphism

Let $(C, d_0)$ be some cochain complex over commutative ring $R$ and $R$ is an algebra over a field $k$, then I could consider complex $(C[t], d_0+td_1)$ over ring $R \otimes k[t]$ where $d_1$ is some ...

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249 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 ...

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### Obstruction map for local singularities via tangent (Andre-Quillen) cohomology

Let $R$ be a local singularity (for example $R=\mathbb{C}[[x_1, \ldots , x_n]]/I$) ring over $\mathbb{C}$. Let $\mathbb{L}_{R}$ be a cotangent complex of $R$, then one can define tangent ...

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### 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 ...

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### 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 ...

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### What's the relation between the heat kernel proof of the index theorem and deformation quantization?

In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat ...

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### Is Nijenhuis–Richardson bracket a BV bracket?

Let $g$ be a finite dimensional Lie algebra, and let me denote $A=(\bigwedge g^* \otimes g, d)$ the Chevalley-Eilenberg complex that calculates cohomology of the Lie algebra with coefficients in the ...

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### 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 ...

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### Proof of Hensel's lemma by using the deformation theory

I am thinking about the simplest version of Hensel's lemma. Fix a prime $p$. Let $f(x)\in \mathbf{Z}[x]$ be a polynomial. Assume there exists $a_0\in \mathbf{F}_p$ such that $f(a_0)=0\mod p$, and ...

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### 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 ...

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### 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 ...

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### 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 ...

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### Formal series convergence in deformation quantization and $C^*$-condition

A link between formal series convergence in deformation quantization (strict deformation quantization) and producing $C^*$-algebras instead of mere $*$-algebras (which ...

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### 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 ...

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### Existence of logarithmic structures and d-semistability

I am reading a paper ( Kawamata, Y.; Namikawa, Y. Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. math. 1994, 118, 395–409.) I have a ...

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### Questions (related to deformation theory?) about modular ideals in mod ell Hecke algebras

I suspect that I'm asking (familiar?) questions from deformation theory in a different language. But I'm an illiterate in deformation theory language; if my suspicion is correct I'd be grateful for an ...

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### Kodaira Spencer map and versal deformation

First I want to clarify what I mean by the Kodaira-Spencer map.
Let's have a family of deformations $\pi:\mathcal{X}\rightarrow B$ of a complex manifold $X=\mathcal{X}_0:=f^{-1}(0)$ (by that I mean ...

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### When does the categorical definition of a module work?

$\DeclareMathOperator{\ab}{Ab}\DeclareMathOperator{\qcoh}{QCoh}$
This entry in the nlab shows that for $A$ a (commutative unital) ring, the category $\mathsf{Mod}_A$ of $A$-modules is equivalent to ...

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### fpqc, formal smoothness

Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field),
let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...

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### formal smooth morphism with a formal smooth source

Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field).
We suppose that X is formally smooth and f is formally smooth and surjective.
Do we have that $Y$ is formally smooth?
Or if it's ...

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### Strict applications of deformation theory in which to dip one's toe

I hesitate to ask a question like this, but I really have tried finding answers to this question on my own and seemed to come up short. I readily admit this is due to my ignorance of algebraic ...

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### functor of Artinian rings in Deformation theory

$k$ : algebraically closed field
$\mathcal{C}$: category of local Artinian $k$-algebras with residue field $k$
$\hat{\mathcal{C}}$: category of complete local $k$-algebras with residue field $k$
...

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### What is the genus of the limit of a family of singular curves?

Let $\mathcal{X}$ be a flat family of (proper) algebraic curves. If generic fibers in $\mathcal{X}$ are non-singular of genus $g$, then the geometric genus (i.e. genus of the desingularizations) of ...

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### How to determine “genericness” of an element of a family of algebraic varieties?

Given a (flat) family of complex algebraic varieties $X_t$ (say parametrized by $\mathbb{C}$) and a specific $t_0$, how does one proceed to check if $X_{t_0}$ is a 'generic element'?
More precisely, ...

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### calculate T^i-functor for a example

I just start to study Hartshorne's 'Deformation theory'.
I have a stupid question about $T^i$-functors. It may be an easy algebra.
Exercise 1.3.3.
Let $B=k[x,y]/(x^2,xy,y^2)$. Show that ...

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### Uniqueness of deformation family,

Say I want to construct a flat family of affine schemes using deformation theory. To be more specific, suppose I have a local $k$-algebra $A$, and I want to find a flat family $\mathcal{X} \to T$, ...

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### Tangent space in Algebraic geometry and Differential geometry

We know in differential geometry, given a $C^k$ manifold for $k>1$, the tangent space at a point in this manifold is parametrized by curves passing through this point modulo certain equivalence ...

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### Morphism with non-reduced special fibre

Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible ...

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### How do I check whether an orbifold admits deformations?

(Cross-post from math.stackexchange, where it has received no attention.)
Orbifolds $\mathbb{C}^2/\mathbb{Z}_n$, given by the action $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ with $\zeta$ a primitive ...

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### Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem?

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define
...

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### Non-proper intersection of projective schemes

Let $X, Y$ be projective varieties in $\mathbb{P}^n$ for $n>10$. Assume that dimensions of $X,Y$ are greater than $n/2$. My first question is as follows:
Is there any criterion ...

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### Deformation of rational points in a family

Let $\mathcal{X} \to B$ be a family of smooth projective varieties over a field $K$ (possibly finite). Assume that each fiber $\mathcal{X}_b$ of the family has a $K$-rational point. Fix a pair ...