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2
votes
0answers
150 views

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 ...
4
votes
0answers
142 views

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 ...
20
votes
2answers
828 views

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 ...
2
votes
0answers
216 views

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 $k$ ...
4
votes
1answer
231 views

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 ...
3
votes
1answer
260 views

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, ...
1
vote
1answer
172 views

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 ...
0
votes
0answers
270 views

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 $\mathcal{X} \to T$, ...
5
votes
6answers
1k views

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 ...
2
votes
1answer
444 views

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 ...
1
vote
0answers
85 views

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 ...
6
votes
1answer
233 views

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 ...
1
vote
0answers
113 views

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 ...
0
votes
0answers
155 views

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 ...
4
votes
1answer
349 views

Complete intersection space curves

Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ ...
2
votes
0answers
110 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 ...
4
votes
4answers
313 views

Non-Drinfeld--Jimbo Deformations and Finite Quantum Groups

As is well-known, for the compact semi-simple Lie groups $G$, there exist non-commutative Hopf algebra deformations ${\cal O}_q[G]$ of their coordinate algebras ${\cal O}[G]$, the so-called ...
4
votes
1answer
176 views

Is there a rigid curve in a product of complex manifolds?

Let $X=Y\times Z$ be a product of complex manifolds $Y,Z$. Is it true that there exists no rigid curve on $X$? Here I mean by a rigid curve a curve which is not a member of any family of curves on ...
4
votes
0answers
112 views

Deformation rings and change of group

Let $f : G' \subset G$ be an injection between profinite groups such that $G'$ is normal in $G$ (typical situation which I deal with : $G$ the absolute Galois group of a local field, $G'$ an open ...
2
votes
1answer
179 views

cohomology of restrictions of vector bundles to deformations

Suppose $X \subset Y$ is a pair of varieties, and $s \in H^0(N_{X/Y})$ is a section. This corresponds to a first-order deformation $X' \subset Y \times \text{Spec}(\mathbb{C}[\epsilon]/\epsilon^2)$ of ...
1
vote
0answers
89 views

Koszul alg deformations

Is it known the maximal class of Koszul algebras for which any deformation is Koszul?
4
votes
1answer
266 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
1answer
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
1answer
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
2answers
245 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
1answer
389 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
1answer
115 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
1answer
198 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
1answer
191 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
3answers
325 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
2answers
552 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 ...
12
votes
1answer
1k 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
1answer
218 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
2answers
254 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
1answer
98 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
1answer
99 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
1answer
199 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
0answers
195 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
2answers
583 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
0answers
167 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
1answer
457 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
1answer
506 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
0answers
149 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
0answers
133 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
1answer
249 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
3answers
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
0answers
252 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
0answers
381 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
1answer
195 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
1answer
286 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, ...