Questions tagged [ag.algebraic-geometry]
for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
0
votes
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
votes
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
votes
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
vote
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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 ...
