All Questions
22,770 questions
5
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226
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Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{...
- 985
1
vote
0
answers
116
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When is a fiberwise-very-ample line bundle on a fibered surface also $k$-very ample?
Let $g : Y\rightarrow\text{Spec }k$ be a smooth proper curve, and let $f : X\rightarrow Y$ be a family of stable curves. Consider the line bundle $\mathcal{L} := \omega^{\otimes 3}_{X/Y}$. It's known ...
- 7,527
3
votes
0
answers
176
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Étale morphisms of derived schemes and stacks
Conventions: In the below,
unless otherwise stated, terms regarding derived algebraic geometry will follow the conventions of Yaylali.
an algebraic stack will be a stack $\mathscr{S}$ over a base ...
- 1,349
7
votes
0
answers
202
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Finite generation and finite presentation over a truncated valuation ring: is there an easier proof?
Let $K^+$ be a valuation ring which is $\pi$-adically complete for some pseudouniformizer $\pi$.
Nagata 053E proved that every finitely generated and flat (equivalently, torsion-free) $K^+$-algebra is ...
- 16.1k
9
votes
3
answers
696
views
I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
1
vote
1
answer
237
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Generalization of Gromov-Witten theory counting surfaces, 3-folds, etc
I don't work on the Gromov-Witten theory, but I find that I need to study the following problem, which seems to be similar to the Gromov-Witten theory:
Let $X$ be a variety and $\alpha_{1}, \cdots, \...
- 305
5
votes
1
answer
362
views
Unramified fppf cohomology
Let $F$ be a number field, $\mathcal{O}_F$ its ring of integers and $G$ an fppf $\mathcal{O}_F$-group scheme.
See the question Unramified Galois cohomology of number fields for unramified cohomology ...
- 265
3
votes
1
answer
159
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Reference request: generalized Jacobian variety for higher dimensional variety
Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
- 389
2
votes
0
answers
194
views
Zariski Connectedness Theorem in Complex Geometry
Let $f: X \to Y$ be a proper surjective morphism of complex irreducible varieties such that general fibre of $f$ is connected and $Y$ integrally closed\normal. Say, we even assume wlog $Y=\text{Spec}(...
- 5,998
4
votes
0
answers
175
views
What is the equivalent of Artin gluing for quasicoherent sheaves?
Given a topological space or locale $X$ and an open $j : U \hookrightarrow X$ with closed complement $i : K \hookrightarrow X$, the inverse image functor $\langle i^*, j^* \rangle : \textbf{Sh} (X) \...
- 15.9k
4
votes
0
answers
148
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'Naive cotangent complex' as 1-truncation of cotangent complex
In the stacks project, there is a 'naive version' of cotangent complex $NL_{S/R}$ for a ring morphism $R\rightarrow S$, given by the chan complex $(I/I^{2}\rightarrow \Omega_{R[S]/R}\otimes_{R[S]}S)$ ...
- 618
4
votes
1
answer
296
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Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it ...
- 335
3
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0
answers
197
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Concrete reasons to study derived categories of quasi-coherent sheaves on algebraic stacks?
Title says it all. There seems to be a lot of work done on derived categories of quasi-coherent sheaves on stacks, and yet I couldn't find many "external" applications. I'm mainly wondering ...
- 143
1
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0
answers
78
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Computing Chow groups of affine, simplicial toric varieties
Let $k$ be an algebraically closed field. Let $X$ be an $n$-dimensional affine, simplicial toric variety over $k$. There exists an $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$ such that $...
- 639
6
votes
1
answer
306
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Hochschild cohomology and differential operators
The Hochschild-Kostant-Rosenberg theorem says, that for a commutative algebra $R$ over a field $k$ with certain smoothness and finiteness, we have an identification $\mathrm{HH}^\bullet(R)\cong \...
- 985
2
votes
0
answers
155
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Symmetric powers for a short exact sequence of vector bundles
If $0 \to A \to B \to C \to 0$ is a short exact sequence of vector bundles on $X$, is it necessarily true that $[{\rm Sym}^k B] = \sum_{i=0}^k [{\rm Sym}^i A \otimes {\rm Sym}^{k-i} C]$ in the ...
- 2,964
3
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0
answers
147
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Kodaira vanishing + simple connectedness implies Fano
To avoid monkey business, let $X$ be a smooth complex projective variety of dimension $m$. As usual, $K_X$ is its canonical line bundle. Let us further assume that $X$ is simply-connected and the ...
- 12.3k
4
votes
1
answer
220
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Does every cubic threefold contain a genus 5 curve of degree 8?
Since a genus $5$ curve $C$ of degree $8$ is a complete intersection of $3$ quadrics $Q_1,Q_2,Q_3$ in $\mathbb{P}^4$, I would guess that $C$ is contained in a cubic threefold $X = \mathbb{V}(f)$ when ...
- 679
0
votes
0
answers
98
views
Differential of the evaluation map of the Kontsevich moduli space
Let $X$ be a smooth projective variety, and $\beta$ a curve class on it. We have the Kontsevich moduli space $\overline{\mathcal M}_g(X, \beta)$ of stable maps from genus $g$ curves to $X$ with class $...
- 19
1
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0
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104
views
Reference about the semiabelian variety associated to a stable curve
If a stable curve $C$ of genus $g$ is such that its dual graph has Betti number $1$, then the Jacobian of $C$ is a semiabelian variety of torus rank $1$. It is described by Mumford that the data of a ...
24
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1
answer
861
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The congruence subgroup property for mapping class groups and a conjecture of Grothendieck
This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.
Recall that a subgroup $K$ of a ...
- 25k
2
votes
0
answers
103
views
Representability of stack of finite maps between curves
I am interested in the following moduli problem: The moduli functor $\mathcal{F}$ has $T$-points:
a nodal $n$-pointed curve $C/T$ of genus $g$.
a nodal $b$-pointed curve $D/T$ of genus $h$.
a finite ...
- 223
2
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0
answers
153
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A schematic representability of an algebraic space with group action
In the book "Néron Models" (BLR), there is a statement as follows (on page 164):
Let $S$ be a locally noetherian scheme and let $G$ be a smooth algebraic group space over $S$ with connected ...
- 291
0
votes
0
answers
88
views
Geometry of prym locus
The celebrated solution to the Schottky problem provides a beautiful geometric characterization of Jacobians among all principally polarized abelian varieties (ppavs). One might hope for a similarly ...
4
votes
0
answers
227
views
Smoothness of complex analytic subspaces
Say I have a complex analytic subspace $X$ of a complex manifold. Additionally:
$X$ is a topological manifold, and
For each $x \in X$, the set of derivatives at $x$ of smooth paths holomorphic discs ...
- 121
2
votes
0
answers
88
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Conjecture on ordinary reductions of smooth complex projective varieties and Its context
I am interested in the conjecture suggesting that many reductions of a smooth complex projective variety are ordinary, as mentioned in Remark 5.1 of the paper by Mustaţă and Srinivas:
Ordinary ...
2
votes
1
answer
159
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Complexification of Néron models of Abelian varieties
Let $A$ be an abelian variety of dimension $g$ over the quotient field $K$ of a DVR $R$ which is a subfield of the complex field $\mathbb{C}$. Then by a result of Grothendieck, we know that there is a ...
- 85
2
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0
answers
95
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Confusion about the Lefschetz standard conjectures for abelian varieties in the integral setting
Let $(A,\theta)$ be a principally polarized abelian variety of dimension $d$ over a number field $k$. By the hard Lefschetz theorem
$$H^2(A,\mathbb{Q}_{\ell}) \xrightarrow{\theta^{d-2}} H^{2d-2}(A,\...
- 679
4
votes
1
answer
510
views
Help with understanding a rigid geometry proof
I am trying to read Coleman's paper "$p$-adic Banach Spaces and Families of Modular Forms". In Proposition A5.5, he considers a quasi-finite morphism $f:Z\to Y=\mathbb{B}^1_K$ where $Z$ is a ...
- 141
2
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0
answers
245
views
Does automorphism of classifying stack come from automorphism of group?
Let $k$ be a field and $G$ be a group. For simplicity, let us assume $\operatorname{char}k=0$ and $G$ is a finite abelian group. We do not assume $k$ is algebraically closed. If there is an ...
- 253
1
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0
answers
87
views
Birational geometry of special divisor varieties and double covers of curves [closed]
Let $\pi: \tilde{C} \to C$ be an étale double cover of a smooth non-hyperelliptic curve $C$. Associated to this cover is a principally polarized abelian variety $(P, \Xi)$, called the Prym variety, ...
user538396
2
votes
0
answers
58
views
$L^2$ approximation of delta functions on real algebraic varieties and asymptotic bounds
Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Consider a probability measure $\mu$ on $X(\mathbb{R})$, absolutely continuous with respect to the Lebesgue measure induced ...
1
vote
0
answers
87
views
Equidistribution of Frobenius Classes
Let $G$ be a reductive group over $\mathbb{Q}$. Let $K$ be a maximal compact subgroup of $G(\mathbb{C})$. Let $S$ be a finite set of primes. For each prime $p$ not in $S$, let $Frob_p$ be a conjugacy ...
- 11
5
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0
answers
145
views
Symmetric groups acting on rational surfaces
Let $X$ be a complex projective rational surface. Is there an upper bound on $n\in\mathbb{N}$ such that $S_n\subset \text{Aut}(X)$? Here $S_n$ is the symmetric group on $n$ elements.
- 193
2
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0
answers
168
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When do we have $\bigoplus_{i + j = n} R^i f_* \mathbb{Q}_\ell \otimes_{\mathbb{Q}_\ell} R^j g_* \mathbb{Q}_\ell \cong R^{i + j} h_* \mathbb{Q}_\ell$?
Milne, Étale Cohomology, theorem 8.5 states the following version of the Künneth formula (in slightly greater generality). Let $\Lambda$ be a finite commutative ring. Let $X, Y, S$ be schemes with $S$ ...
- 531
3
votes
2
answers
283
views
Definition of $M_{1,0}$
Is there an explicit construction of the moduli space $M_{1,0}/\mathbb{Q}$ of genus $1$ curves whose set of $R$-points, for a $\mathbb{Q}$-algebra $R$, is the set of isomorphism classes of genus $1$-...
- 2,907
3
votes
0
answers
122
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A Zariski's Main Theorem for affine morphisms
Let $f: X\to Y$ be a birational affine surjective morphism with geometrically connected fibers between smooth $\mathbf{C}$-varieties.
Question: Is $f$ an isomorphism?
If $f$ is proper, then $f$ is ...
- 389
3
votes
2
answers
271
views
Orbits under the automorphism group of projective space
Let $\mathbb{P}^d_K$ be projective space of dimension $d\geq 1$ over an infinite field $K$. Let $x\in\mathbb{P}^d_K$ with $\dim\overline{\lbrace x\rbrace}=n\leq d-1$.
My question: is the set $\lbrace ...
- 133
3
votes
1
answer
408
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Twists of elliptic curves
I have a few questions regarding twists of elliptic curves.
In the context of the Shafarevich group, I see people refer to the group of twists of an elliptic curve $E/\mathbb{Q}$ by $H^1(\mathbb{Q}, ...
- 2,907
5
votes
1
answer
264
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Non-existence of power divided structure on a maximal ideal of truncated polynomial rings (example from Koblitz)
In 3.3 of Berthelot-Ogus's book Notes on Crystalline Cohomology, an example from Koblitz is exhibited without proof.
Let $k$ be a field (or a commutative ring) with characteristic $p>0$, $$A:=k[x_1,...
- 441
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
2
votes
1
answer
306
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Serre functors and global dimensions
Let $k$ be a field.
Let $\mathcal{C}$ be an abelian category (over $k$).
We say that $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that
$$
\operatorname{Ext}^i(M,...
- 889
1
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1
answer
141
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Why is $2A_0(X)=0$ for a cubic threefold $X$ containing a line, over an arbitrary field $k$
I can't quite follow Proposition $2.1$ of "UNIVERSAL UNRAMIFIED COHOMOLOGY OF CUBIC FOURFOLDS CONTAINING A PLANE". I posted this on Math stackexchange but got no answer.
Let $X$ be a smooth ...
- 679
1
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0
answers
92
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Counting conjugacy classes of completely reducible subgroups in general linear groups [closed]
Let $G = GL(n, q)$ be the general linear group over the finite field $\mathbb{F}_q$ with $q$ elements, where $q$ is a power of a prime $p$. Let $m$ be a positive integer dividing $q-1$. Suppose $\...
7
votes
0
answers
124
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Projections of closed geodesics under the modular function
In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...
- 68.8k
5
votes
0
answers
175
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Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
- 641
1
vote
0
answers
84
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Describing monoidal categories of positive-weight representations geometrically
Let $G=\mathbb{G}_m.$ The monoidal category $\mathcal{C}=\text{Rep}(G)$ of $G$-representations (also known as the category $\text{Gr}$ of graded vector spaces) can be written geometrically as $\...
- 423
1
vote
1
answer
127
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Vanishing of higher morphisms for pair moduli
Consider a moduli scheme $P$ of sheaves $F$ on a Calabi-Yau n-fold $X$ with a section $s\in H^0(F)$. Alternatively, such objects can be described as maps $\mathcal{O}_X \xrightarrow{s} F$ called pairs....
- 988
6
votes
0
answers
338
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Deeper meaning behind the occurrence of the factor $\frac{\log q}{i}$ in Deninger's results
In two papers Deninger proved the following:
If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$,...
- 1,805
1
vote
1
answer
160
views
Is every log resolution a sequence of blowups?
Suppose we have a variety $X$ over a field of characteristic zero. Choose any ideal sheaf $\mathcal{I}$ on $X$. Is every log resolution of the pair $(X,\mathcal{I})$ a sequence of blow ups? I cannot ...
- 11