Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
7,837
questions with no upvoted or accepted answers
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Calabi-Yau structure on cotangent bundle?
Let $M$ be a smooth (compact) manifold, my question is when the cotangent bundle $T^*M$ has a Calabi-Yau structure.
Certain constructions are known, for instance, if $M=\Sigma\times S^1$ or $M$ is a ...
- 763
4
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140
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Scaling-Invariant Orbits of Semisimple Group Representations
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Note that if $V$ is the adjoint representation of $G$, then ...
- 4,870
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201
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Deformations and moduli of semistable sheaves in mixed characteristic
Let $X$ be a projective scheme over an algebraically closed field $k$. There is the coarse moduli space $M_X$ parametrizing semistable sheaves on $X$ with fixed reduced Hilbert polynomial $p$. Now, ...
- 203
4
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Shimura varieties and Maximal conditions
Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for ...
- 2,181
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675
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Ample Line Bundles on Algebraic Spaces
The sources known to me (Knutson's Algebraic Spaces and Pascual-Gainza's Ampleness criteria for algebraic spaces) define a line bundle $L$ on an algebraic space $X$ (over a base scheme $S$) to be ...
- 11.9k
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Can one use the equivariant Thom-Gysin sequence on a singular affine variety?
Let $X$ be a smooth complex affine variety. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite stratification of $X$ into smooth locally closed subvarieties. Let $T$ be a complex algebraic torus ...
- 4,870
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465
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Points of moduli space of semistable sheaves and S-equivalence classes
Let $X$ be smooth projective and connected curve over an algebraically closed field $k$. One knows the description of $k$-valued points of the moduli space $M_X$ of semistable vector bundles of fixed ...
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4
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determine if a toric variety is Gorenstein
Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod End(V_{\omega_{i}})\times\prod\...
- 3,432
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942
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Questions about dessin d'enfants, trees and their Shabat polynomials
This will be a series of questions, a few of which have been troubling me for quite a while now. Before I jump right in, let me first introduce a few notions which I will assume.
(Note: All of these ...
- 829
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377
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Fibers of flat morphism
Let $f:X \to Y$ be a surjective flat projective morphism between Noetherian schemes. Assume that $Y$ is non-reduced. Denote by $g:X \times_Y Y_{red} \to Y_{red}$ the pull-back of $f$ by the natural ...
- 1,573
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229
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Non-reduced flag Hilbert schemes
Let $P, Q$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Assume that $pr_2(Hilb_{Q,P})$ is positive dimensional where $pr_2$ is the natural projection map onto the second coordinate and $Hilb_{Q,...
- 1,020
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489
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Intuition about Toroidal Embeddings
I have been trying to understand the very basics of toroidal embeddings, and the definitions on the face of them are not terribly daunting. I've been going with the "locally analytically looks like a ...
- 437
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Categories of sheaves and Kan Extensions
This is quite a broad question regarding constructions of categories of sheaves in geometry.
Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...
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355
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Stack theoretic image?
If $X \xrightarrow{f} Y$ is a morphism of schemes then the scheme theoretic image of $f$ is the smallest closed subscheme $Z \subset Y$ through which $f$ factors through.
Is this notion defined for ...
- 3,938
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506
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Discrete valuations for which Abhyankar inequality is strict
The background to my question, in a nutshell, is: If $k$ is a field and $X$ a $k$-variety, i.e. an integral, separated, finite type $k$-scheme, which discrete rank $1$ valuations on $k(X)$ come from ...
- 4,400
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Algorithm for solutions to quadratic forms over number fields
Are there any know (preferably implemented) algorithms to find solutions to quadratic forms over number fields (or global fields)?
I am especially interested in the quaternary case. There exist some ...
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Is there a way to compute explicitly global sections of tangent sheaf to a projective hypersurface?
Let $f\in \mathbb{C}[z_{0},\ldots,z_{n}]$ be an irreducible homogeneous polynomial and $X=V(f)\subset\mathbb{P}^{n}$ the projective hypesurface associated. I want to find (if any) explicit generators ...
- 1,707
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fields are $K(\pi,1)$
I want to prove that fields are (étale) $K(\pi,1)$.
One possibility would be to apply "If $Y \to X$ is a connected pro-étale Galois cover such that $Y$ is $K(\pi,1)$, then so is $X$." to $\mathrm{...
user19475
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The support of a finite type module on an algebraic space
I'd like to ask this question to make sure I understand a very basic thing about supports. Let $X$ be an algebraic space and F a quasi-coherent sheaf on it of finite type.
In here the schematic ...
- 1,265
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what does the decomposition theorem say for a Lefschetz pencil?
The setting is the following: let $X$ be a smooth projective variety (say over $\mathbb{C}$), $D$ a simple normal crossings divisor on $X$ and $(H_t)_{t \in \mathbb{P}^1}$ a Lefschetz pencil on $X$ ...
- 41
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Homotopy-theoretic measure of operations on sheaves failing to be sheaves
Here's something I've been wondering about for a few weeks:
Consider a topological space $X$ and a sheaf of rings $\mathscr O_X$ on $X$. Suppose $\mathscr{F}$ and $\mathscr{G}$ are $\mathscr O_X$ ...
user1437
4
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289
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Is Normalization of log smooth scheme smooth?
Let $f:Y\rightarrow X$ be a finite flat morphism between smooth schemes over $Spec k$, where $k$ is a perfect field. Let $D$ be an irreducible and smooth divisor of $X$, $U=X\setminus D$ the ...
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597
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When are Abelian schemes projective?
Under what conditions on the base $X$ are Abelian schemes $\mathcal{A}/X$ projective, and projective in which sense?
user19475
4
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296
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Which orbits of a separable representation of the infinite unitary group are closed?
Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all ...
- 965
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777
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Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
- 165
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299
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Clifford index of curves on a surface
Suppose C is a smooth curve lying on a smooth surface X such that C belongs to |nH|, where H is an ample divisor on X. How to calculate the Clifford index Cliff(C) ?
- 101
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365
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Closed subschemes and the analytification functor
Let $X$ be a scheme of finite type over $Spec(\mathbb C)$ and let $i:Y \hookrightarrow X$ be a closed subscheme, defined by a sheaf of ideals $J \subset \mathcal O_X$.
Then there is an induced map $i^...
- 41
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435
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trivial deformation of a smooth affine scheme over complete DVR
Let $X$ be a affine smooth scheme finite type over $A/pA$, where $A$ a complete DVR and $chk=p>0$.
I know that since $H^{2}=0=H^{1}$, we have a unique lifting to $A/p^{2}$. In algebraic schemes ...
- 1,911
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141
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cyclic covers over a non-algebraically closed field
This is a question about chapter 3 of Esnault-Viehweg "Lectures on vanishing theorems".
Let $k$ be an algebraically closed field. Let $X$ be a smooth, projective variety over $k$ and
$$
D=\sum a_j ...
- 41
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403
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When $R/(f)$ is regular?
For R being a commutative regular excellent Noetherian ring of finite Krull dimension which conditions on $f\in R$ can ensure that the ring $R/(f)$ is regular (so, I want a sufficient condition)? I do ...
- 16.5k
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2k
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exactness on pullback of sheaves
Suppose $i:Y⊂X$ is a closed subvariety of a variety X.
Suppose there is an exact sequence of coherent sheaves on X:
$$
0→A→B→i∗C→0
$$
Suppose C is locally free on Y. Pulling the above exact sequence ...
- 41
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321
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"Reductive Groups and Hilbert Schemes" - Reference
Bezrukavnikov and Ginzburg have unpublished notes, 'Hilbert Schemes and Reductive Groups' (referenced here, for example): does anyone know what became of these notes? Did Bezrukavnikov-Ginzburg ...
- 1,181
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What is the structure of the stack of complexes supported in dimension less than r?
Let $X$ be something. (smooth and projective variety over C are my assumptions)
The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...
- 1,265
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Homogeneity degree one functions of a matrix argument
I am interested in homogeneity degree one (scalar-valued) functions of a matrix argument. The simplest setup is as follows. Let $X$ be a symmetric $3\times 3$ matrix with real entries. Let $f$ be a ...
- 153
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stratification by gerbes of a symmetric power of a gerbe
We work over $\mathbb{C}$.
I'm trying to understand the following result (a lemma from the Stacks project), in some particular example.
The Lemma says that for an algebraic stack $X$ for which the ...
- 867
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472
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Tangent space of the moduli stack of Drinfeld modules
I am going through the proof of Thm 1.5.1 of Laumon, Cohomology of Drinfeld modular varieties, which says that a certain map of stacks is smooth. To prove this, Laumon considers the tangent space of a ...
- 12k
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Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field
The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.
A variety $X$ over a finite field $k$ is liftable if there ...
- 381
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Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree
Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.
Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?
I want to exclude finite etale ...
- 381
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sign in the polarization of Hodge structures
I have been trying to understand signs and conventions in the Hodge theory. Clearly, I am no expert in this area. I apologize if I ask stupid question(s) and make wrong comment(s).
In Deligne's ...
- 41
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Classify cross-sections of the adjoint quotient for a semisimple algebraic group?
[This question arises from trying to understand an incompletely formulated earlier question
here.]
Let $G$ be a semisimple algebraic group over an algebraically closed field $K$ of
characteristic $p \...
- 52.4k
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Does the Albanese map satisfy Torelli's theorem
Let $M_h$ be the moduli space of canonically polarized varieties with Hilbert polynomial $h$. Let $M_h \to A_g$ be the Albanese map, with $g$ an integer which depends on $h$ and $A_g$ the moduli space ...
- 381
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97
views
Fibers of the secant map
Let $X\subset CP^N$ be a homogeneous (or perhaps just smooth) complex
subvariety and let $S^r(X)$ denote its abstract $r$-th secant variety
(the incidence variety in $X\times \cdots \times X\times CP^...
- 835
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312
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Invariants of reductive group actions and completion
I'm trying to understand the extent to which taking invariants of a reductive group action "commutes" with completion. More precisely:
Let $X = \operatorname{Spec} A$ be a reduced finite type affine ...
- 533
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Is there an arithmetic analogue of Drinfeld's count of a number of 2d irreps of fundamental group of a curve ?
There is a paper by V. Drinfeld 1981, which title is "Number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field".
It gives a formula for this ...
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440
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Cohomology of a sheaf with only one stalk
Let $X$ be a proper scheme over a henselian discrete valation ring. I have a Nisnevich sheaf $F$ of which has only one stalk at the generic point of $X$ (and all other stalks vanish).
I believe that ...
- 1,413
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384
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Reference request: virtual fundamental class of moduli of stable maps
Let $f:U\longrightarrow \overline{M}_{0,n}(\mathbb{P}^m,d)$ is the universal family with morphism $\pi:U\longrightarrow\mathbb{P}^m$ and let $X\subset\mathbb{P}^m$ be a hypersurface defined by a ...
- 227
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410
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Applications of moduli of curves theory
Are there some applications of moduli of curves theory? I was wondering if moduli of curves theory is used (or could be used) for doing research in applied mathematics.
I am doing my PhD in algebraic ...
4
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386
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A question on an intuitive way to look at stacks
I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...
- 805
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234
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Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds
Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection
$$ \...
- 39.3k
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545
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Singular fibers of an elliptic fibered K3 surface.
Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
- 41