The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
0answers
112 views

Base change and geometrically generic reduced fiber

Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic ...
1
vote
0answers
120 views

Deformation of a Hopf algebra

A deformation of a Hopf algebra is defined as follows. On page 171 of the book a guide to quantum groups, Remark 2, it is said that Any deformation of a Hopf algebra $A$ as a bialgebra is ...
1
vote
1answer
85 views

Existence of curve nodal at given set of points

I am reading this (https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/Univcodim2invent.pdf) paper of Voisin, but I am having some trouble with the proof of Sublemma 2.8 (it might be something very ...
7
votes
0answers
223 views

Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$ $$\require{AMScd} \begin{CD} M^D ...
2
votes
1answer
155 views

Families of smooth projective varieties over dvr

Let $R$ be a discrete valuation ring with residue field $k$, an algebraically closed field of characteristic zero and $\pi:X\to \mbox{spec}(R)$ a smooth, projective family of surfaces. Denote by ...
4
votes
1answer
141 views

Deformation long exact sequence of GW theory in the analytical setting

Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume ...
2
votes
1answer
75 views

Torsion free sheaves in flat families

Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that ...
1
vote
1answer
264 views

Local deformations of Lagrangian submanifolds in holomorphic symplectic manifold and their intersections

Let $Y\subset X$ be a Lagrangian submanifold in a holomorphic symplectic manifold $X$. We know that there exists a local moduli space $M$, which parametrizes lagrangian submanifolds in $X$(there are ...
6
votes
0answers
129 views

Deformation of Complex Spaces

I am trying to learn about deformations of complex spaces from the paper of Palamodov. I am particularly interested in the relative tangent cohomology. Is there any other modern reference to this ...
1
vote
1answer
170 views

Tangent complex of dgla/ twisted dgla

I am looking for a theorem which says if we twist an $L_\infty$ quasi-isomorphism between dgla's by a Maurer-Cartan element, we'll get a new $L_\infty$ quasi-isomorphism. Let $(\mathfrak{g}, d, ...
0
votes
1answer
185 views

Euler characteristic on flat families of quasi-projective schemes

Let $A$ be a noetherian integral domain (may be regular). Let $\pi:X \to \mathrm{Spec}(A)$ be a flat morphism. Suppose that each fiber of $\pi$ are quasi-projective. Let $\mathcal{F}$ be a coherent ...
4
votes
0answers
123 views

Deformations of the moduli space of ppav's

Consider the complex algebraic moduli space $X:=\mathcal A_g^n$ of ppav's of dimension $g$ with some high enough level $n$ structure (so that it represents the corresponding functor). Can one compute ...
1
vote
1answer
280 views

Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
11
votes
1answer
361 views

Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
2
votes
0answers
113 views

Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$

Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...
1
vote
0answers
82 views

Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
2
votes
1answer
153 views

p-adic modular forms, Hecke algebra, deformation theory and modular curves.

Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation ...
4
votes
0answers
117 views

Obstruction to lifting of global sections of invertible sheaves

Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an ...
0
votes
1answer
85 views

Pencil of singular affine hypersurfaces differing only in the constant term

Question--quick version: Does there exist a (nonconstant) polynomial $f \in \mathbb C[x,y,z]$ such that for all $c \in \mathbb C$, the affine hypersurface cut out by $f + c$ is singular? Motivated ...
11
votes
0answers
189 views

Squeezing physics out of formal deformation quantizations

I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...
5
votes
2answers
315 views

Automorphisms and infinitesimal deformations of a smooth complete intersection

Let $X\subset\mathbb{P}^{n+c}$ be a smooth complete intersection of dimension $n$. Is it known when $Aut(X)$ is finite ? Does there exist a formula for the dimension of the tangent space to the ...
4
votes
1answer
143 views

A moduli problem inspired by Stein factorization

Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ ...
1
vote
0answers
103 views

A naive question on rational equivalence of varieties

Let $X$ be a projective scheme and $\pi:\mathcal{Z} \to \mathbb{P}^1$ a surjective morphism of finite type such that for any pair $t_0, t_1 \in \mathbb{P}^1$, the fibers $\mathcal{Z}_{t_0}$ and ...
4
votes
1answer
182 views

When is the flatness locus non-empty

Let $k$ be an algebraically closed field, $f:X \to Y$ be a surjective proper $k$-morphism locally of finite presentation between irreducible noetherian schemes. Assume that $Y$ is reduced. Under what ...
5
votes
1answer
157 views

Is the Quot-scheme over non-singular curve reduced

Let $k$ be an algebraically closed field, $C$ a non-singular projective curve over $k$ of genus at least $2$ and $\mathcal{F}$ a locally free sheaf on $C$. Let $r,d$ be two integers satisfying ...
6
votes
1answer
116 views

Does the degeneracy of the Frölicher spectral sequence vary in families?

I would like to know if there are any known examples of families of complex manifolds for which the Frölicher spectral sequence of one fibre degenerates on the $E_m$ page and the spectral sequence of ...
2
votes
0answers
66 views

quasi-ordinary singularities on a versal deformation?

Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities. ...
7
votes
0answers
171 views

Lefschetz morphisms from the relative tangent sheaf exact sequence?

Let $X\subseteq {\mathbb{P}}^N$ be an $n$-dimensional complex projective manifold. Denote by $\pi\colon U\to X$ the affine cone of $X$ with the vertex removed; it is a $\mathbb{C}^*$-bundle over $X$. ...
1
vote
0answers
131 views

On tangent space of relative quot schemes in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic and $f:X \to S$ be a smooth, flat, projective morphism between noetherian $k$-schemes. Assume that $S$ is a non-singular ...
1
vote
0answers
73 views

Normal bundle of reducible nodal curve

Let $X=\mathbb P^3_k$ ($k=\bar k$) and $l_1,l_2,l_3$ three distinct lines such that $l_1\cap l_2\neq \emptyset$,$l_2\cap l_3\neq \emptyset$, $l_1\cap l_3=\emptyset$ and $l_2\cap l_3\neq l_1\cap l_2$ ...
3
votes
0answers
191 views

Deformation of finite coverings between smooth projective varieties

Assume that we have a finite covering $$f \colon X \longrightarrow Y,$$ where $X$ and $Y$ are smooth, complex projective varieties of dimension $n$. Therefore we obtain a splitting $$f_* \mathscr{O}_X ...
1
vote
0answers
191 views

Quantities associated to deformed sheaves

I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...
8
votes
2answers
383 views

Inverse Galois problem for $GL_2$ of a compact local ring

Let $A$ be complete noetherian local ring with maximal ideal $m$ and residue field $A/m$ a finite field (in other words, $A$ is a noetherian compact local ring) For which $A$ as above is there a ...
3
votes
0answers
68 views

Derived Deformations of associative algebras

Let $k$ be a field (if necessary for my question, we can assume its characteristic to be zero). In the non derived context, we can then define deformations of an associative algebra $S$ as follows: ...
6
votes
1answer
293 views

Isotrivial families with non-zero Kodaira spencer map

Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...
1
vote
0answers
203 views

Flat cohomology of an ordinary liftable Calabi-Yau threefold

Let $k$ be a perfect field of characteristic $p>0$ and consider an ordinary liftable Calabi-Yau threefold $X_{0}/k$. By this I mean that $H^{i}(X_{0},B_{X_{0}/k}^{j})=0$ for all $i\geq 0$ and ...
3
votes
0answers
174 views

What does “control of a deformation problem” mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
2
votes
1answer
179 views

Relative dualizing sheaf (reference, behavior)

Let $\mathcal{C}\rightarrow S$ a flat projective family of locally complete intersection projective curves over a integral noetherian scheme (say a spectrum of a local ring). I was wondering whether ...
4
votes
1answer
191 views

Infinitesimal deformations of fake projective planes (or ball quotients)

This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial. By ...
4
votes
2answers
346 views

Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers. Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...
0
votes
1answer
172 views

Model over DVR for smooth projective curves

Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...
1
vote
1answer
137 views

The canonical bundle of an infinitesimal deformation

Let $X_0$ be a smooth projective variety over the complex numbers and let $X$ be an infinitesimal deformation of $X_0$ over the ring of dual numbers. If the canonical bundle of $X_0$ is ample (resp. ...
5
votes
2answers
409 views

Specialisation of rigid varieties

Recall that a variety $X$ over a field $k$ is called rigid if $H^1(X, T_X) = 0$. I am interested in understanding this property under specialisation. Let $R$ be a discrete valuation ring and let ...
8
votes
1answer
415 views

Deformations of Ext rings

Let $k$ be a base ring and $k[x]$ the ring of polynomials in an indeterminate $x$ over $k$. Consider a (not necessarily commutative) algebra $A$ over $k[x]$ and two $A$-modules $M$ and $N$. Then for ...
6
votes
1answer
215 views

Infinitesimal deformations of a singular projective surface

Let $X$ be a normal projective surface with just two singular points $x_1,x_2\in X$, where $X$ has rational quotient singularities. Assume that both the singularities in $x_1$ and in $x_2$ admit a ...
1
vote
1answer
226 views

Strong form of Grothendieck's algebrization theorem

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field $k$ ($R$ is not necessarily complete), $X$ a regular surface over $\mathrm{Spec}(R)$ and a sequence of locally ...
2
votes
1answer
227 views

When is the Hom-scheme connected?

Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor ...
10
votes
2answers
355 views

Infinitesimal deformations of the formal group of $\mathbb{G}_m$

For a commutative ring $R$, consider the formal group $\widehat{\mathbb{G}}_m$ over $R$ that is the completion of $\mathbb{G}_{m, R}$ along its identity section (naively, $\widehat{\mathbb{G}}_m$ is ...
2
votes
1answer
105 views

formally etale deformations of algebras

Let $A$ be a local artinian ring with residue field $k$, $S$ a $k$-algebra. Suppose there is a formally etale deformation $B$ of $S$ over $A$, i.e. a flat $A$-algebra $B$ such that $S\cong ...
5
votes
2answers
417 views

References for the moduli space of complex structures

I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...