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1
vote
1answer
167 views

Smoothness and smoothness over formal neighborhood

Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$. We assume that the map on the level of fomal neighborhoods $X_{x}\rightarrow Y_{y}$ is formally smooth, can we find ...
7
votes
0answers
164 views

Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
1
vote
1answer
102 views

On the infinitesimal lifting property of non-singular affine schemes

Let $k$ be an algebraically closed field (not necessarily of characteristic $0$), $X$ a non-singular affine closed subscheme in $\mathbb{A}^n_k$ for some $n \ge 2$. Denote by $I_X$ the ideal of $X$ in ...
0
votes
1answer
195 views

Line on a hyper surface

Assume $X$ is a hyper surface in $\mathbb{P}^n$, can one always find a closed immersion $i:\mathbb{P}^1 \rightarrow X$?
2
votes
1answer
140 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 ...
1
vote
0answers
87 views

On the universal property of certain representable functors and rational sections

Let $P_1,P_2$ be two Hilbert polynomials of subschemes in $\mathbb{P}^n$. Denote by $H_{P_1,P_2}$ the corresponding flag Hilbert scheme (parametrizing pairs $(X\subset Y)$ where $X$ has Hilbert ...
1
vote
1answer
126 views

An application of the Grauert's upper semi-continuity theorem

Let $X$ be a smooth projective variety, $A$ a complete discrete valuation ring, $Y=\mbox{Spec} A$ and $f:X \to Y$ a smooth, projective, surjective morphism. Denote by $y$ the closed point of $Y$. Let ...
1
vote
2answers
122 views

Jacobian of a curve and field extension

Let $K$ be a field of characteristic zero and $X_K$ a smooth projective curve on $K$. Denote by $\bar{K}$ the algebraic closure of $K$ and $X_{\bar{K}}$ the base change of $X_K$ to $\bar{K}$. Under ...
3
votes
2answers
138 views

Injectivity under flat base change of the Picard group on smooth projective curves

Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...
4
votes
0answers
69 views

$q$-deformations of fundamental equation of information and entropies

Classical information theory: fundamental equation of information In classical information theory, the information $I(A)$ of an event $A$ (any element of the $\sigma$-algebra $\mathcal F$ of a ...
4
votes
2answers
391 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 ...
4
votes
0answers
139 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
1answer
199 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 ...
5
votes
3answers
113 views

graded generalization of the Moyal–Weyl product

Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?
2
votes
1answer
102 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
0answers
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 ...
9
votes
1answer
265 views

Motivation behind the definition of hochschild cohomology

For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as ...
4
votes
1answer
179 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 ...
6
votes
1answer
523 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 ...
4
votes
1answer
150 views

Simple maps: Flat versus locally trivial

In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term ...
13
votes
1answer
308 views

Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory

Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...
1
vote
0answers
53 views

Differential graded Lie algebras and gauge equivalent

For the Kodaira-Spencer complex $\Omega^{0,*}(T^{1,0})[[t]]$ on a compact complex manifold, with a Hermitian metric. It is well known that finding formal series solution $\Xi = \Xi_1 t^1 + \Xi_2 t^2 + ...
5
votes
1answer
204 views

formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian ...
0
votes
1answer
143 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 ...
1
vote
0answers
108 views

Hochschild cohomology and bar resolutions

I asked the following question on mathstack but didn't receive any comments, so I thought I'd try my luck here. Let $A$ be an associative algebra over a field $k$. One can define $HH^n (A,A)$ as $ ...
4
votes
1answer
165 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
1answer
119 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
1answer
68 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
1answer
278 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
1answer
152 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} ...
2
votes
1answer
86 views

Equivalence of star products on two differents Poisson algebras?

Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
4
votes
0answers
255 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
1answer
237 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
1answer
145 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. ...
8
votes
1answer
263 views

Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded Lie algebra" as explained first in ...
3
votes
0answers
114 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
0answers
250 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 ...
6
votes
2answers
362 views

Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone. In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion. Let's fix a representation $\bar{\rho}$ of a group ...
7
votes
1answer
220 views

alternative to Kontsevich formality

Has anyone considered an alternative approach to Kontsevich formality in which the DGCA of poly-vector fields is deformed to an $L_\infty$-algebra?
2
votes
1answer
136 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
0answers
105 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
0answers
170 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
1answer
168 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
0answers
212 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
1answer
365 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
1answer
134 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
1answer
139 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
1answer
216 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
1answer
169 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
0answers
106 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$\}$, ...