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### Some examples of $\mathbb Q$-Gorenstein smoothing

I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition. Definition. For a normal projective surface $X$ with quotient ...
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### Surjectivity of the Kodaira-Spencer map

Let $X$ be a complex projective manifold. Let $B$ be the closed subscheme of $H^1(X,T_X)$ defined by $\mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal defining the origin. In other words, $B$ is a ...
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### Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?

Main Question: What Is the correpondence between flows and vector fields in algebraic geometry? Here is a more precise statement could be an answer If it was true (I have no idea it is): "...
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### Cohomology of tangent sheaf of a singular hypersurface

Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s. Does there exists a formula, perhaps in ...
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I learnt that $\mathbb{P}^1 \times \mathbb{P}^1$ is rigid, but can be deformed to a non-rigid Hirzebruch surface $S$. Suppose $\pi: M \to B$ is such deformation such that $\mathbb{P}^1 \times \mathbb{... 1answer 217 views ### Normal bundle to fibers of a rational morphism Let$f:X\dashrightarrow C$be a rational fibration from a 3-dimensional variety$X$to a curve$C$such that generic fiber is smooth and different fibers intrsect in smooth curves. Take$S$to be a ... 0answers 108 views ### Maurer-Cartan equation for Lie groups/homogeneous space vs. Maurer-Cartan of deformation theory What is the relationship between the Maurer-Cartan equation $$d\theta + \dfrac{1}{2}[\theta,\theta] = 0$$ satisfied by Maurer-Cartan forms on Lie groups, or by pullbacks of Maurer-Cartan forms along ... 0answers 256 views ### Restriction of a global moduli functor that admits a coarse moduli space Let$F:(Sch/k)^{o}\to Sets$be a functor, where$Sch/k$is the category of schemes over a field$k$. Suppose that$F$admits a coarse moduli space, let it be$M$. Consider a$k$-point$x\in M$(which ... 1answer 191 views ### Deformation Quantization I am a beginner and I want to learn about deformation quantization. Please suggest me with which book or notes, I should start? 0answers 127 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 ... 0answers 140 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 ... 1answer 92 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 ... 1answer 305 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 @>... 1answer 167 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 X_0... 1answer 192 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 C=(\... 1answer 88 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 ... 1answer 279 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 ... 0answers 138 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 ... 1answer 228 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, [-,-]... 1answer 204 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 ... 0answers 130 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 ... 1answer 313 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 ... 1answer 366 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 ... 0answers 114 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. ... 0answers 89 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 ... 1answer 180 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 \bar{\... 0answers 120 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 ... 1answer 90 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 ... 0answers 208 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 ... 2answers 345 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 ... 1answer 144 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 ... 0answers 105 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 \... 1answer 196 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 ... 1answer 167 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 \... 1answer 125 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 ... 0answers 67 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. ... 0answers 179 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. ... 0answers 143 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 quasi-... 0answers 77 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 (... 0answers 193 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 ... 0answers 192 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 ... 2answers 386 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 ... 0answers 70 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: ... 1answer 304 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 ... 0answers 211 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$j\...
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 ...