All Questions
22,548 questions
3
votes
0
answers
166
views
Étale descent of étale motives for algebraic spaces
Let $X$ be a (sufficiently nice) algebraic space, one can define the category of étale motives $\mathbf{DA}(X,R)$ (with $R$ a ring) like the case of schemes (see for instance, La réalisation étale et ...
- 893
0
votes
2
answers
282
views
Can a variety be the graph of a function in more than one way?
Let $V\subset \Bbb R^n$ be an irreducible affine variety of degree $\ge 2$ and $U_V\subseteq V$ a (Euclidean) open subset. Suppose that $U_V$ is the graph of a rational function, that is, there is an ...
- 13.6k
3
votes
0
answers
122
views
Weil restriction of a bunch of points or more general disjoint unions
$\DeclareMathOperator\Spec{Spec}$For a finite extension of fields $k'/k$, let $R_{k'/k}$ denote the Weil restriction functor from quasiprojective $k'$-schemes to quasiprojective $k$-schemes, defined ...
- 472
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
1
vote
0
answers
114
views
Simultaneous elimination of variables in multiple polynomials
I have a system of $n=O(1)$ non-homogeneous polynomials of total degree $d=O(1)$ $p_1,\dots,p_r\in \mathbb Z[x_1,\dots,x_n]$. I would like to eliminate $n-1$ variables simultaneously from the $n$ ...
- 13.9k
2
votes
0
answers
137
views
Irreducible non-reductive subgroup in GL(n) over a characteristic 0 field
Let $V$ be an $n$-dimensional vector space over a characteristic $0$ field $k$ (or better, let $V=\mathbb{A}^n_k$). I wonder whether the following is true:
Absolutely irreducible subgroups $H$ of $\...
- 455
5
votes
1
answer
237
views
Methods of finding integer solutions beyond the reach of direct search
Consider a classical problem: given a polynomial Diophantine equation $P(x_1,\dots,x_n)=0$, determine whether it has an integer solution. While this problem is undecidable in general, we may still ...
- 7,182
2
votes
1
answer
331
views
Completion of a local ring is noetherian (under some hypothesis)
I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly:
Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
- 293
1
vote
0
answers
68
views
Uniqueness of a canonical homography decomposition
Consider a multi-camera system with $n \geq 3$ calibrated cameras, each represented by a projection matrix $P_i \in \mathbb{R}^{3 \times 4}$ for $i=1, \dots, n$. We first want to detect and track ...
4
votes
1
answer
418
views
Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
- 41
0
votes
1
answer
302
views
Is a bijective regular map between affine varieties a homeomorphism?
Let $X \subset \mathbb{A}^n,~ Y \subset \mathbb{A}^m$ be affine varieties. Consider a regular map $f: X \to Y$. If $f$ is bijective, can we conclude that $f$ is an open mapping w.r.t the Zariski ...
2
votes
0
answers
237
views
Obscure action of derivations on group schemes (SGA 3 Exp III)
In what follows, I will refer to prop. 0.8 in SGA 3 Exp. III which can be found for example at the link (https://webusers.imj-prg.fr/~patrick.polo/SGA3/). I will quickly introduce the notation without ...
- 21
4
votes
2
answers
585
views
Krull dimension in non-algebraically closed fields
Let $K$ be a field (not algebraically closed) and $F$ be its algebraic closure.
Let $X \subseteq K^n$ be Zariski closed, and $Y$ be the Zariski closure of $X$ inside $F^n$.
Is it true that $\dim(X) = \...
5
votes
0
answers
226
views
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
views
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 ...
- 7,537
3
votes
0
answers
179
views
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
203
views
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 ...
- 16.2k
9
votes
3
answers
699
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
238
views
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
365
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
views
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
196
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}(...
- 6,048
4
votes
0
answers
177
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
149
views
'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
297
views
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
votes
0
answers
198
views
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
vote
0
answers
80
views
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
307
views
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
157
views
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 ...
- 2,974
4
votes
0
answers
150
views
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 ...
- 12.4k
4
votes
1
answer
221
views
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
vote
0
answers
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
votes
1
answer
869
views
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
votes
0
answers
154
views
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
views
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
views
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
votes
0
answers
96
views
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
votes
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
vote
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
votes
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
votes
0
answers
168
views
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
284
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
views
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