All Questions
22,770 questions
1
vote
0
answers
125
views
When is a vector bundle on a Shimura variety an automorphic vector bundle?
Let $(G, X)$ be a Shimura datum, let $K \subset G(\mathbb{A}_f)$ be an open compact subgroup, and denote by $\text{Sh}_K(G,X)$ the Shimura variety whose complex points are given by $G(\mathbb{Q})\...
- 159
3
votes
1
answer
286
views
Derived Koszul complex
Let $X$ be a projective variety over $\mathbb{C}$ and $V$ be a vector bundle over $X$. Let $\pi: V\to X$ be the natural projection.
Let $i: X\to V$ be the zero section map. Let $V^\vee$ be the dual ...
- 653
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
2
votes
4
answers
212
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 157
3
votes
1
answer
231
views
Are principal parabolic group scheme bundles Zariski locally trivial?
Let $P$ denote a parabolic subgroup scheme of $\operatorname{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes ...
- 2,907
4
votes
1
answer
242
views
On the degeneration of the elliptic surface $E(n)$
The following matter should be widely known (if true). I am sorry for my ignorance!
For the natural $n$, let $E(n)$ be the corresponding elliptic surface.
In the analytic world, there exists a well-...
- 153
5
votes
1
answer
883
views
Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
- 1,093
11
votes
0
answers
183
views
Are algebras with rational structures dense in varieties of real Lie nilpotent algebras?
One says that a real nilpotent Lie algebra has $\mathbb Q$-structure if it has a basis with rational structure constants. It is well known that there are nilpotent Lie algebras without $\mathbb Q$-...
- 723
2
votes
0
answers
91
views
Adelic description of moduli of stable vector bundle of rank n (over finite fields)?
Let $Bun_G$ be the moduli stack of $G$-bundles on a (geometrically irreducible smooth projective) curve $C$ over a finite field $k$, where $G$ is a split reductive group over $k$. Since Weil, we know ...
- 6,612
0
votes
0
answers
98
views
$h^0(X, 4H-5E)$ on weak Fano threefold
Let $X$ be a smooth weak fano threesfold arising as the blowup of a smooth curve $C$ of degree and genus $(d,g)=(10,2)$ in a rank 1 smooth fano threefold $Y$, $-K_Y^3 = 22$. Let $H$ be a hyperplane in ...
- 71
0
votes
1
answer
119
views
Detecting singular points from a parametrization
Suppose $r(t)$ parametrizes some, say algebraic, curve in the plane. It can certainly be that $r$ is smooth but the curve is not, since $r$ resolves double points by passing through them at different ...
- 13
2
votes
0
answers
153
views
Uniqueness and existence of maps
I am currently reading the Berkeley lectures on Perfectoid Spaces by Scholze and Weinstein. In the section "The adic open unit disk over $\mathbb{Z}_p$" we encounter from Proposition 4.2.6 ...
0
votes
0
answers
60
views
The generalized Laplace expansion for tensor
I'm reading this paper https://arxiv.org/abs/1308.3860.
In the Appendix (page 22), the author uses a generalized Laplace expansion for the determinant tensor, as shown in the picture1.
But I only ...
- 1
2
votes
1
answer
200
views
Ampleness of the pullback of the relative dualizing sheaf of $\overline{\mathcal{M}_{g,n+1}}\rightarrow\overline{\mathcal{M}_{g,n}}$
There is a natural map $f : \overline{\mathcal{M}_{g,n+1}}\rightarrow\overline{\mathcal{M}_{g,n}}$ identifying the source with the universal family over the target. Let $\sigma_1,\ldots,\sigma_n$ be ...
- 10.7k
1
vote
0
answers
64
views
Existence of a special uniformizer along a smooth section of a prestable curve
Let $R$ be a complete DVR with fraction field $K$, uniformizer $\pi$ and alg. closed residue field $k$.
Let $f : X\rightarrow \text{Spec }R$ be a prestable model of $\mathbb{P}^1_K$ with a $R$-section ...
- 7,527
5
votes
1
answer
290
views
Why does a line bundle on an abelian variety give a group extension only if it is algebraically trivial?
If $X$ is an abelian variety and $L$ is a line bundle, deleting the zero-section one obtains a diagram
$$
0 \to \mathbb G_m \to Y \stackrel{\phi}{\to} X \to 0
$$
where $\mathbb G_m = \phi^{-1}(0)$,
...
1
vote
1
answer
106
views
Iterated optimal transport
Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
- 13
5
votes
1
answer
260
views
Principal bundles over smooth projective curve
Let $X$ be a smooth and connected projective curve over $\mathbb{C}$ and $G$ a reductive connected group over $\mathbb{C}$. Fix a faithful representation $G \subseteq \mathrm{GL}_n$.
Given a $G$-...
- 1,061
1
vote
1
answer
139
views
Specialization of points on the generic fiber in a prestable model of $\mathbb{P}^1$
Let $R$ be a complete DVR with uniformizer $\pi$, fraction field $K$ and residue field $k$. We assume $k$ is algebraically closed.
Let $X$ be a prestable model of $\mathbb{P}^1_K$ over $R$, so $X$ is ...
- 7,527
5
votes
2
answers
358
views
Canonical conics pulling back to polynomials on rational normal curve
(In following all schemes are formed over $\Bbb C$)
Let $C:=\nu_d(\Bbb P^1)$ the rational normal curve obtained via $d$-folded Veronese map $\nu_d: \Bbb P^1 \to \Bbb P^d$. The quadrics on $\Bbb P^d$ ...
- 5,998
0
votes
0
answers
163
views
Free action of finite group on a scheme
Let $X$ be an affine scheme over $S$ and let $G$ be a finite group acting freely on $X$.
I saw two definitions in the literature regarding "free action", the first that the map $G\times_S X\...
- 119
2
votes
0
answers
60
views
Birational change the variety to the higher model if the nonKLT locus is connected?
I was reading the paper BCHM, there is an application of BCHM results to the proof of inversion of adjunction in this paper:
Corollary 1.4.5 (Inversion of adjunction). Let $(X, \Delta)$ be a log pair ...
- 225
5
votes
0
answers
216
views
Lifting a morphism between quasi-projective varieties
Let $\mathcal{V}$ be an affine algebraic variety over $\mathbb{R}$, $G$ be a finite group acting freely on $\mathcal{V}$. Consider the quotient space $Y:=\mathcal{V}/G$, which itself is a quasi-...
- 364
5
votes
1
answer
290
views
Compatibility of natural transformations in a six-functor formalism
Suppose we are given a six-functor formalism and a cartesian diagram
$$\require{AMScd} \begin{CD} X @>\tilde{g}>> Z \\ @V \tilde{f} V V @V Vf V \\ Y @>g>> W\end{CD} \,.$$
There are ...
- 913
7
votes
0
answers
249
views
Phantoms and Geometry
Let $\mathcal{D}(X)$ be the bounded derived category of coherent sheaves on a smooth projective variety $X$. An autoequivalence $\Phi: \mathcal{D}(X) \to \mathcal{D}(X)$ is called phantom if it ...
- 71
3
votes
1
answer
326
views
Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 31
1
vote
0
answers
167
views
Hypergeometric sheaves on $\mathbb{A}^{1}_{E}$
Let $m, n$ be non-negative integers. Assume that $\boldsymbol{\chi} = \left( \chi_i \right)_{1 \leq i \leq m}$ and $\boldsymbol{\eta} = \left( \eta_j \right)_{1 \leq j \leq n}$ are two collections of ...
- 21
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,964
2
votes
0
answers
162
views
Equivariant Künneth formula for partial flag variety
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$. Let $P$ be a parabolic subgroup of $G$, $\mathscr{F}:=G/P$ the partial flag variety associated to $P$. For a $G$-variety $X$, ...
- 653
4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
2
votes
1
answer
399
views
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 2,964
8
votes
0
answers
401
views
Langlands program in higher dimensions
We can view the Langlands program in each of its versions (local/global, arithmetic/geometric) as giving a description of the finite-dimensional representations of the étale fundamental group of a ...
- 91
5
votes
1
answer
567
views
Dualizing sheaf of nodal curve
Let $C$ be a connected, nodal (I'm working with definition from Alper's notes on Stacks & Moduli, see p 210), projective curve over an alg closed field $k$, beeing everywhere smooth except at a ...
- 5,998
4
votes
1
answer
164
views
Describing the compactified Jacobian of a nodal curve
$\DeclareMathOperator{\Pic}{Pic}$Let $C$ be an integral projective curve over $\mathbb C$, which is smooth except for a single node $x\in C$. Let $M$ be the moduli space of stable torsion-free sheaves ...
- 1,286
3
votes
0
answers
389
views
Status of motives in higher category theory: motives and algebraic cycles through a higher categorical perspective
A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives.
Primary question:
Have there been any recent developments/advances on the above question? If not,...
0
votes
0
answers
97
views
Weight space decomposition of smooth representation of complex algebraic torus
Question: Let $T=(\mathbb{C}^{*})^{n}$ and $\pi:\mathcal{E}\to \mathbb{C}^{n}$ a smooth complex $T$-equivariant vector bundle (i.e. $\pi$ is $T$-equivariant and $T$ acts on the fibers linearly). The ...
- 101
2
votes
1
answer
154
views
$R^1\Gamma = 0$, and the Mumford stability
Let $S$ be a smooth projective surface with an ample divisor $H$ so that $K_S \cdot H < 0$.
Let us consider the Mumford slope $\mu(E) = \frac{H \cdot c_1(E)}{\operatorname{rk}(E)}$ on $\...
4
votes
1
answer
252
views
Multiplicative cancellation for trivial vector bundles
Let $X$ be a scheme, ${\mathscr L}$ an invertible ${\mathscr O}_X$-module, and $d$ a positive integer. If ${\mathscr L}^{\oplus d} \simeq {\mathscr O}_X^{\oplus d}$, does it follow that ${\mathscr L} \...
- 318
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 ...
- 883
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
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
135
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
235
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
327
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
234
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
584
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) = \...
