All Questions
22,546 questions
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Sequence of MMP with scaling cannot be isomorphism
Let $(X,B)$ be a projective klt pair, H is an $\mathbb{R}$-divisor s.t. $K_X+B+H$ is nef. Suppose that we can run $(K_X+B)$-MMP with scaling $H$(that is, the flip exists) and denote $\alpha_i:X \...
- 11
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0
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87
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finiteness of quotient map
I am new to algebraic space s and stacks. My question is the following:
Let $X$ be a scheme and $G$ be a group scheme acting on $X$. We have the natural map $\pi : X \to X/G$. When $\pi$ will be a ...
- 613
1
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0
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85
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Projection from a point and singularity
Let $X \subset \mathbb{P}^n$ be a hypersurface with $n \ge 3$. Let $x \in X$ be a closed point. Consider the map given by projection from $x$:
$$\phi: X \dashrightarrow \mathbb{P}^{n-1}$$
Suppose that ...
- 1,060
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82
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Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
- 785
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0
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41
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Descend local system to the canonical model of Shimura varieties
Suppose $(G,X)$ is a Shimura data, and $E$ be its reflex field. In page 33 of this paper, it constructs an etale local system on the canonical model $\mathrm{Sh}(G,X)_{K,E}$ (variety over $E$) for any ...
- 785
4
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2
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165
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Connectedness of degeneracy loci
Let $E, F$ be vector bundles of ranks $e, f$ on a smooth variety $X$ of dimension $n$.
Let $\varphi : E \to F$ be a morphism and let $k$ be such that $n > (e-k)(f-k)$.
Fulton-Lazarsfeld's theorem ...
- 4,111
4
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171
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Intuition on geometry of sections
Premise: I have asked the same question on math.stackexchange about a week ago, without receiving answer. Therefore I've decided to ask it also here. If this violates the rules, I apologize and I'll ...
2
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96
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Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The ...
- 6,622
3
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1
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160
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Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
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4
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Stable points in GIT: geometric picture
Is there a geometric picture justifying why "stable points" in GIT (Geometric Invariant Theory) are actually called "stable"? Stable, with respect to which effect? (Here, I have ...
- 12.4k
6
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3
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2k
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Complex projective space as a $U(1)$ quotient
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}$As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $\U(1)$, ...
- 12.9k
3
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111
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Bertini's theorem at a fixed point
Recently, I am learning Bertini's theorem because I encounter "generic smooth" problem during my research. I'm not an algebraic geometer and I read the Hartshorne Chapter 3 Theorem 10.8 to ...
- 12.9k
3
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105
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Jacobian of a reducible curve with arbitrary singularities
Let X be a reduced, reducible curve over $\mathbb{C}$ with locally planar singularities, and let $\widetilde{X}$ be its normalization. I am interested in the Jacobian varieties $\mathrm{Jac}(X)$ and $\...
- 31
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96
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Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
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267
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Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
- 63.9k
23
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2
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2k
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Locales as geometric objects
There is the following analogy:
$$\begin{array}{cc} \text{frames} & - & \text{commutative rings} \\ | && | \\\text{locales} & - & \text{affines schemes}\end{array}$$
Here, ...
- 42.4k
6
votes
1
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282
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effective descent of coherent sheaves
I am new to stacks and algebraic spaces. I have the following question:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...
- 388
2
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0
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125
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Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 63.9k
4
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88
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Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$.
The paper "Reductive ...
- 6,622
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134
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Universal semistable curve
For the moduli of stable curve, we have the result by Deligne-Mumford-Knudsen that there's an isomorphism of moduli spaces
$$\Phi : \overline{\mathcal{X}}_{g,n} \xrightarrow{\sim} \overline{\mathcal{M}...
- 834
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2
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197
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What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 103
1
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182
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"Reflex field" for $\mathbb H/\Gamma$ for $\Gamma$ non-congruence
Suppose $\Gamma$ is a non-congruence arithmetic subgroup of $PGL_2(\mathbb Z)$, and $\mathbb H$ is the upper half plane of $\mathbb C$. Then by Belyi's theorem we know $\mathbb H/\Gamma$ is an ...
- 785
4
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1
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445
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Exact functor in syntomic cohomology
By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site.
Is it also true for a finite flat ...
CommunityBot
- 1
3
votes
1
answer
367
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Variants of Grothendieck section conjecture
Let $X$ be a smooth projective variety defined over a field $k$. We fix the following notations : $\overline{k}$ denotes the algebraic closure of the field $k$, $X_{\overline{k}}$ denotes the variety $...
- 156
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1
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201
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Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 12.9k
3
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1
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299
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Motives and birational invariance
One can construct non-isomorphic smooth projective varieties which define the same motive by blowing up $\mathbb{P}^2$ at five points. I think I learned this here at MathOverflow. But these examples ...
- 11
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1
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215
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Geodesic flows and Killing fields
How well-known is the following result:
Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ ...
- 4,787
5
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1
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278
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Set-theoretic generation by circuit polynomials
Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
- 1,046
4
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1
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183
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About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local fields
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be an elliptic curve defined over $K$. Tate's algorithm can be used to compute the Kodaira symbol of the reduction type of $E$.
However, I ...
- 37k
2
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0
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120
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Analogs of Plücker relations in Clifford algebras, and Bott periodicity (?)
Classical Plücker relations can be viewed as conditions on coefficients of an element $x=\sum_Sc_Se_S$, $S=(i_1,...,i_k)$, $i_1<\cdots<i_k$, $\{i_1,...,i_k\}\subset\{1,...,n\}$ of an exterior ...
- 18.4k
3
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2
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366
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Rational divisors on Calabi–Yau threefolds
Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are ...
- 317
16
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1
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688
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Approximating zero sets of real polynomials with "less complicated" polynomials
Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...
CommunityBot
- 1
2
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1
answer
172
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Actual zeros of tropical Laurent polynomial
I consider the tropical semi-ring $(\mathbb{R},\oplus,\odot)=(\mathbb{R},\max,+)$. I know that the tropiclisation of any (Laurent) polynomial $p\in\mathbb{R}[x_1^{\pm1},...,x_n^{\pm1}]$ given a ...
CommunityBot
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3
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1
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138
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Can the coefficients of a Taylor series be expressed as rational functions for an affine variety?
Let $k$ be an algebraically closed field and $V \subseteq k^n$ an affine variety corresponding to a prime ideal $P \subseteq k[t_1, \dots, t_n]$. For $x\in V$ let $O_x = \{p/q \mid p,q\in k[t_1, \dots,...
- 5,359
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108
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L.c.i locus of Hilbert scheme of points on singular varieties
Let $X$ be an algebraic variety over $\mathbb{C}$. What can we say about the l.c.i. locus of $\text{Hilb}^n(X)$?
When $X$ is smooth, it is well-known that the l.c.i. locus of $\text{Hilb}^n(X)$ is ...
- 385
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0
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240
views
What do we do when $G$ doesn't have a Shimura variety?
Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the ...
- 12.9k
2
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1
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270
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Jacobian fibration of elliptic fibration: basic relations between Enriques invariants
Let $f: X \to B$ be an elliptic fibration, so proper map from smooth surface $X$ onto smooth conn. curve over alg closed base field $k$ with connected fibers such that almost all fibers are elliptic ...
- 246
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0
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89
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Extend algebraic morphism to a compactification with normal crossing boundary
Suppose $X$ and $Y$ are smooth algebraic variety over a char $0$ field $k$, and $f:X\to Y$ a morphism. I want to ask whether there exists compactifications $\bar X$ and $\bar Y$ such that $\bar X\...
- 785
2
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1
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127
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Changing the weight space for an eigenvariety
Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing ...
- 37k
3
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1
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505
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Conservativity of stalks of ind-constructible sheaves
I have a simple question about the conservativity of stalks of ind-constructible sheaves. Let $X$ be a topologically noetherian scheme, $S$ a set of geometric points of $X$ corresponding bijectively ...
CommunityBot
- 1
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0
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151
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Compactification of the Jacobian of singular curves via parabolic modules
I would like to better understand a certain compactification of the Jacobian variety of a singular algebraic plane curve as described in Cook's Ph.D. 1993 thesis Local and Global Aspects
of the Module ...
- 35.5k
1
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0
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92
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Compactification of smooth varieties with normal crossing boundary
I see in this paper, page 46, the second sentence of 4.1, that every smooth variety over a characteristic $0$ field can be embedded into a proper smooth variety with normal crossing boundary, and the ...
- 785
2
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1
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185
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Number of rational points of a quotient of connected linear algebraic groups
Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius ...
- 23
3
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1
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133
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Is a simply connected locally 2-connected complex a union of spheres and planes?
Let $X$ be a (potentially infinite) 2-dimensional simplicial complex. Then each link at a vertex $x\in X$ is a graph.
Question. If $X$ is simply connected and each link is 2-connected (in the sense ...
- 12.3k
4
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173
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Why are the Hodge filtrations on cohomology canonically bounded?
If $X$ is a complex projective variety of dimension $n$ then the de Rham cohomology $H^{k}(X,\mathbb Q)$ naturally has a mixed Hodge structure with an increasing weight filtration $W_\bullet$ and a ...
- 141
3
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1
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151
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Locally nilpotent derivations and triangularizability
If $ k $ is a field of characteristic zero and $ \delta \in T_{\mathbb{A}^{n}_{k}/k} $, then $ \delta $ is triangular if $ \delta = \sum_{i=2}^{n} f_{i}(x_{1},\dots,x_{i-1}) \frac{\partial}{\partial ...
- 816
4
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1
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245
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Group action on affine variety induces faithful action on tangent space
I have a queestion about the proof of Lemma 2.2 from the paper arxiv 1105.3739:
Let $G$ be a group acting faithfully on an irreducible affine variety $X=\operatorname{Spec}(A)$ over $k= \Bbb C$. ...
- 12.9k
2
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0
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98
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An injective map in equivariant algebraic K-theory
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent ...
- 12.9k
18
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2
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1k
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Proving algebraicity of compact Riemann surfaces without Chow's theorem
I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
2
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0
answers
138
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Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
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