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Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

0
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0answers
25 views

Is a family of Cohen-Macaulay modules again Cohen-Macaulay (non-noetherian case)

Let $A$ be a local non-noetherian $\mathbb{C}$-algebra, $B$ a finitely generated, regular $\mathbb{C}$-algebra and $M$ a finite $B \otimes_{\mathbb{C}} A$-module, flat over $A$. Suppose that $M \...
3
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0answers
26 views

smoothness of Hurwitz spaces with arbitrary ramification profiles

Fix integers $n\ge 3,d\ge 2$, and partitions $\lambda_1,\ldots,\lambda_n$ of $d$. Let $\mathcal{H}$ be the moduli space of degree $d$ covers $f:C\to\mathbb{P}^1$ that have ramification profiles $\...
5
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0answers
179 views

How should one approach reading Higher Algebra by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT ...
1
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0answers
21 views

Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $C^n$? By homogeneous I mean a quadratic Poisson structure of the form $\{z_i,...
0
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0answers
59 views

Do Plucker relation follow from a subsystem of equations?

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$, \begin{align*} P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} ...
2
votes
1answer
198 views

Is the support of a flat module generically flat?

Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y ...
1
vote
1answer
144 views

Leray spectral sequence from hypercohomology

Context: Deligne, Theorie de Hodge II, section 1.4.8. Let $f:X\rightarrow Y$ be a map between spaces; $\mathcal{F}$ a sheaf of abelian groups on $X$, and $\mathcal{F} \rightarrow \mathcal{F}^{\...
1
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0answers
159 views

About quotient varieties

Let $K$ be a field, $L/K$ be a finite Galois extension with Galois group $G$ such that $(char(K),|G|)=1$ and $K$ contains all $|G|$th roots of unity. Let $B$ be a $L$-algebra of finite type endowed ...
2
votes
0answers
127 views

Examples of endomorphisms of complex curves

I am looking for examples of smooth projective complex curves $X$ of genus at least $2$ and with algebraic endomorphisms $f:X\rightarrow X$ of degree at least $2$. In the case of elliptic curves and $\...
4
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1answer
116 views

$h$ is identity as soon as $h(\Sigma)\cap \Sigma$ contains at least 5 points

In the paper "Normal Subgroups in the Cremona Group", under remark 5.1 they stated that for any generic set $\Sigma \subset \mathbb{P}^2_\mathbb{C}$ of $k$ points, and $h$ is an automorphism of $\...
5
votes
1answer
233 views

How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...
8
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0answers
194 views

Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?

$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons. ...
7
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0answers
111 views

Are maximal tori conjugate Zariski-locally?

Let $S$ be a scheme and let $G\to S$ be a reductive group scheme. Then $G$ admits a maximal torus etale-locally, and any two maximal tori are conjugate etale-locally, by Theorem 3.2.6 and Corollary 3....
2
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1answer
171 views

Restriction of Ext sheaves on closed subschemes

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
-1
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0answers
136 views

On a theorem on the face of Hardy's book [on hold]

I wonder what's the theorem of geometry on the green facepage in the following website: https://www.google.co.jp/search?q=a+course+of+pure+mathematics+by+g.+h.+hardy+geometry&source=lnms&tbm=...
2
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0answers
81 views

A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
2
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0answers
108 views

Can one always find a bigger global resolution

Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-...
2
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0answers
103 views

Restriction of the sheaf of relative differentials

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...
3
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0answers
68 views

Isogeny to a semi-simple group

Let $G$ be a split reductive group over a field of characteristic zero. Is there always an isogeny $G \to H \times T$ with $H$ semi-simple and $T$ a split torus ? I have in mind the case of $\text{GL}...
10
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0answers
187 views

Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
2
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0answers
95 views

Structure of the End group scheme of an abelian scheme over ring of integers

Let $O$ be the integer ring of a p-adic field $K$ (finite extension of $\mathbb Q_p$), $\mathscr{A}$ be an abelian scheme over $S=\operatorname{Spec O}$, consider the group endohomorphism scheme of $\...
0
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1answer
103 views

About a theorem of Mestre

There is a theorem of Mestre witch states: If $K$ is any field, let $p(x) \in k[x]$ is a monic polynomial with degree $2n$, then there exist polynomials $g(x)$ and $r(x)$ with: 1) $g(x)$ and $r(x)$ ...
3
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0answers
107 views

Homotopy colimit description of stacks

Let $F$ be an Artin stack. If $p: X \to F$ is an atlas for $F$, can we express $F$, in the $\infty$-category ${\rm Shv}^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial ...
0
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0answers
161 views

Analysis, topology, algebraic geometry, and number theory interface in a phd thesis [closed]

I am a first year graduate student in mathematics and I was wondering if I could do my phd on a topic which involves analysis, topology, algebraic geometry, and number theory? I am particularly ...
1
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0answers
74 views

Bounding the denominator in the canonical bundle formula

My question concerns with Theorem 3.1 in the paper "A canonical bundle formula" by Fujino and Mori. The theorem claims the following: Suppose $X \to C$ is a fiberation whose general fiber $F$ has ...
6
votes
0answers
108 views

Moduli space of flat connections of Lie group over a 2-torus

We know that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\rm flat}=\mathbb{E} / {S}_N = \mathbb{P}^{N-1} $$ ...
0
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0answers
106 views

Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form $$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$ $$\vdots$$ $$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$ where $h_1(x_1,\dots,x_{...
10
votes
1answer
377 views

Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety. Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$. Let $f : X\to Y$ be a morphism of ...
3
votes
0answers
117 views

Maximum number of integral roots in degree $d$ polynomial?

Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that Each coefficient is bound in absolute value by $B$ Degree of each variable in any monomial is bound by $d$ Total degree is $d'$ $f(x_1,\...
6
votes
2answers
256 views

Outer Hodge groups of rationally connected fibrations

I believe the following is true and well known. Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $\mathbb{C}$. Let $$ f\colon X\rightarrow Y $$ be a surjective map with ...
0
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0answers
102 views

Final step in Coppersmith?

In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
2
votes
1answer
111 views

The centralizer of a semisimple element which is not contained in any proper parabolic subgroups

Let $G$ be a connected, reductive group over a field $k$. Let $A_G$ be the split component of $G$. If necessary, assume $k$ is perfect. Let $g \in G(k)$ be a semisimple element. Then the ...
2
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0answers
88 views

(Singular) metric associated to the higher cohomology

Suppose $X$ is a smooth complex variety and $L$ is a line bundle with a metric $h_L$, then a section $s \in H^0(X, L)$ gives another metric $\tilde h_L:= e^{-\phi}h_L$ where $\phi=\log \|s\|^2_{h_L}$. ...
3
votes
0answers
87 views

When is Tate module of a semiabelian variety over a number field semisimple?

When is the $\ell$-adic Tate module of a semiabelian variety $A$ over a number field $K$ semisimple as a representation of $Gal(K^{alg}/K)$? If $A$ is the product of a torus with an abelian variety, ...
2
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0answers
133 views

Examples of certain compact Kaehler manifolds

A Kaehler manifold is a complex manifold which has a Kaehler metric and Ricci curvature tensor $R_{ij}$. The Ricci curvature tensor is a Hermitian matrix having real eigenvalues. My question is: Is ...
5
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0answers
162 views

Holomorphic structures on vector bundles over $\mathbb C\mathbb P^2$

It is known that every (topological) complex rank $2$ vector bundle over $\mathbb C\mathbb P^2$ admits holomorphic structures. A proof can be found in the book of Okonek, Spindler, Schneider which is ...
1
vote
0answers
144 views

Another question on De Franchis's theorem

Let $X$ be an algebraic curve defined over the complex numbers $\mathbb{C}$ of genus $g > 1$. A theorem of De Franchis states that there exist only finitely many (isomorphism classes of) curves $Y$ ...
3
votes
0answers
122 views

Do profinite completion and homotopy fixed points commute?

Let $X$ be a separated integral normal scheme of finite type over $\mathbb{C}$. It is my understanding that $\mathbb{Z}/2$ acts on the homotopy type of $X(\mathbb{C})$ and its Sullivan 2-profinite ...
0
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0answers
61 views

Representation of symmetric group as Cremona transformations

Question from me and a colleague: Given a matrix \begin{equation} U = \begin{bmatrix} U_{11} & U_{12} \\ U_{21} & U_{22} \end{bmatrix} \quad \text{with } U_{22} \neq 0, \end{equation} ...
2
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0answers
178 views

Are there enough curves (to connect 'points' of f.g. algebras)?

(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the ...
6
votes
0answers
311 views

$\mathbf{Q}_p$ versus $\mathbf{C}$

Let $\sigma_p : \mathbf{Q}_p\to\mathbf{C}$ and $\sigma_{\ell} : \mathbf{Q}_{\ell}\to\mathbf{C}$ be two field homomorphisms, with $\ell\neq p$. Can one describe the compositum field $K$ of $\mathbf{Q}...
3
votes
1answer
137 views

Generic singular hypersurface

I've heard in informal conversations before the claim that: "a generic singular hypersurface has a single singularity of type $\mathbf{A}_1$". What is the precise statement of this result? Where can ...
2
votes
1answer
47 views

Lie-algebra-like relation for totally symmetric 4-tensors

There are many totally symmetric real 4-tensors, $T_{ijkl}$, which satisfy the relation $$T_{ijmn}T_{mnkl} + T_{ikmn}T_{mnjl} + T_{ilmn}T_{mnjk} = c T_{ijkl}$$ with some constant $c$. By the way ...
1
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0answers
73 views

Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
37
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0answers
1k views

What is Prismatic cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...
0
votes
0answers
81 views

Seems like $S$-units equation in algebraic function fields

Let $K/\mathbb F_q$ be a algebraic function field ($q=p^f$), $S$ be a finite set of places of $K$, $O_S$ be the ring of regular functions of $K$ outside $S$ and $a,v\in O^\times_S$, $v\notin\overline{\...
6
votes
0answers
233 views

Explicit examples of Azumaya algebras

I'm trying to understand the Brauer group of a scheme better. I know how to compute $\text{Br}(X)$ as an abstract group in some cases, but don't have a good idea of what the individual Azumaya ...
0
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0answers
89 views

Ample cone generates Picard group? [closed]

Let $X$ be a projective variety over the field of complex numbers. Is it always true that the ample cone generates the Picard group of $X$ ?
2
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0answers
81 views

Variety Isomorphism Problem for Abelian Surfaces

This is a special case of this question, where it is asked whether there exists an algorithm to determine whether two varieties are isomorphic. There, an answer by Bjorn Poonen explains how to solve ...
6
votes
1answer
325 views

Functoriality for $\ell$-adic cohomology - a question

This should a be basic enough question, but I’m a little confused. In proving that $H^*(X,\mathbf{Q}_{\ell})$ is functorial (in the sense of Weil cohomology theories: see axiom D2 here) as $X$ ranges ...