All Questions
22,547 questions
1
vote
1
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249
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Higher cohomology of line bundles and small modifications
I am a PhD student in algebraic geometry and I am blocked on a question that I asked to myself for my research. I am not yet very comfortable with the traditional tools. Also, I apologize in advance ...
- 113
3
votes
0
answers
150
views
$p$-adic points of open subschemes of complete intersections
I'm currently studying the $3\times 3$ magic square of squares problem for a research project. The variety is initially given by the intersection of $8$ quadrics in $\mathbb{P}^8$, but via Gröbner ...
- 31
4
votes
1
answer
284
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Are the two definitions of fppf topology on the category of schemes the same?
Consider the definition of fppf (pre)topology on the category of schemes $\mathrm{Sch}$.
Maybe, most textbooks define an fppf covering of $U\in\mathrm{Sch}$ as a family of morphisms $\mathscr{U} = \{...
- 125
0
votes
0
answers
31
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Zero sets of sums of multivariate polynomials defined recursively with mutually disjoint support
It is difficult in general to say anything about the zero sets of a sum of two or more multivariable polyomials. However, I am interested in two special cases:
I have a family of multivariate ...
- 357
2
votes
1
answer
224
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Example of stable bundle whose pullback is polystable
Kempf (1992): "Pulling back bundles" has the following theorem:
Let $f: Y \rightarrow X$ be a finite morphism. If $\mathscr{W}$ is a bundle on $X$ that is stable with respect to an ample ...
- 577
6
votes
1
answer
293
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When does isomorphism on singular cohomology imply isomorphism on Picard and Brauer groups?
Assume that $f:X\to Y$ is a morphism of complex varieties, and the homomorphisms $H^i_\text{sing}(f)$ are bijective for $0\le i\le 3$ (though possibly $3$ is too much here:)). Under which ...
- 16.9k
3
votes
1
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253
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About decomposition theorem BBD with respect to some stratification
I want to follow up a question from here (how to deduce version 1.a. from version 1).
I know a version of decomposition theorem BBD:
Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex ...
- 133
3
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0
answers
195
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Are local complete intersections of small codimension necessarily (global) complete intersections?
Hartshorne's 1974 conjecture states that a smooth closed subvariety $X$ of $\mathbb{CP}^r$ of dimension $>2/3r$ is necessarily (globally) a complete intersection. Is anything interesting known ...
- 16.9k
2
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0
answers
144
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Picard group of the category of numerical motives
Is anything known about the Picard group of $Chow_{Num}(k, \mathbb{F}_{p})$ (numerical Chow motives with $\mathbb{F}_{p}-$coefficients)?
Perhaps the Picard groups of some other categories of pure ...
- 87
2
votes
1
answer
233
views
existence of a coherent sheaf
I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\...
- 613
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93
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Maximal number of couples of vectors $(r,n)$ such that the product $\langle n_i , n_j\rangle+\langle r_i \times n_i , r_j \times n_j\rangle<0$?
Let $\langle v,w\rangle$ and $v\times w$ stand for the dot product and the cross product of vectors $v,w\in\mathbb R^3$.
Do there exist $2k$ vectors
$$
r_1, \dots, r_k, \; n_1, \dots, n_k\in \mathbb{...
3
votes
1
answer
320
views
Is the Hilbert Mumford Criterion true over the reals?
The Hilbert Mumford Criterion as in Wallach Theorem 3.24 says:
Let $G$ be a linearly reductive subgroup of $GL(n, \mathbb{C})$. Let $(\sigma, V)$ be a regular representation of $G$.
For a vector $v \...
- 161
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0
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85
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Are there algorithms for lowering the degree of the polynomials which generate an ideal by strategically adding new variables?
Suppose I have an ideal $I$ in a polynomial ring $F[x_1,...,x_n]$ generated by a few polynomials of maximum degree $k$.
I want to embed the ideal $I$ into a larger ideal $J$ in a larger ring with ...
- 11
1
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0
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125
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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
292
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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
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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
3
votes
1
answer
231
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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
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243
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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
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183
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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
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91
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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 ...
- 6,622
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votes
0
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98
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$h^0(X, 4H-5E)$ 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
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1
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120
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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
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0
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153
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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 ...
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0
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61
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
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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
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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
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290
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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 $\mathbb G_m = \phi^{-1}(0)$,
...
5
votes
1
answer
261
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
140
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
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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 $\Bbb P^d$ ...
- 6,048
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
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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
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0
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217
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
294
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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
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0
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249
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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
4
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1
answer
328
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 $...
- 41
1
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0
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169
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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 ...
9
votes
2
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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 ...
- 2,974
2
votes
0
answers
163
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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
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0
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267
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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
401
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 $\mathbb C[t,t^{-1}]$...
- 2,974
8
votes
0
answers
402
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
568
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 ...
- 6,048
4
votes
1
answer
165
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
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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 ...
- 893