Questions tagged [ag.algebraic-geometry]
for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
14,707 questions
2
votes
0answers
96 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
93 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
147 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 ...
6
votes
0answers
178 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
148 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
65 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
187 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 ...
3
votes
1answer
147 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
71 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
79 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 ...
38
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
84 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
239 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 ...
2
votes
0answers
85 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
335 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 ...
0
votes
0answers
144 views
Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
8
votes
0answers
240 views
$c_2$ of Calabi-Yau three-folds
Let $Y$ be a smooth compact Calabi-Yau three-fold (over $\mathbb C$, with $\pi_1(Y)=0$). Is it true that $c_2(Y)$ is Poincare dual to an effective curve? If not, can one construct a counter-example?
...
3
votes
0answers
124 views
Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable
I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu.
Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open ...
1
vote
0answers
67 views
Coordinate ring of an equivariant embedding of a homogeneous projective variety
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...
5
votes
0answers
132 views
Does an fppf cohomological class annihilated by an etale cover come from etale cohomological group?
Let $X$ be a scheme, $F$ a sheaf on the fppf site of $X$, and $\alpha\in H^i_{\mathrm{fppf}}(X,F)$ such that it is trivialized by an etale cover of $X$. Does $\alpha$ lie in the image of the canonical ...
1
vote
0answers
123 views
Tangent space to representation variety
In "Varieties of Representations of Finitely generated groups" by Lubotzky and Magid in page vi it claims that
"A. Weil showed that the tangent space to $R_n(T)$ at a representation $\rho$ is a ...
1
vote
0answers
54 views
partially simultaneous resolution of singularity
Let X be a projective manifold and let $D=\sum_{i=1}^{m}a_iD_i\in |mL|$ be an effective divisor on $X$ with SNC support. Let $f:X\to Y$ be a surjective morphism over a projective manifold $Y$. Write $...
5
votes
1answer
232 views
A question on dominant morphism of affine schemes
Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A_{ \mathfrak p}$ is regular local ...
1
vote
0answers
91 views
Boundary of stable maps and transversality
Fulton and pandharipande in FP-notes prove that if we consider $Def_G(\mu)$ as deformation of a map $\mu:C \to X$ from nodal curve with $q$ nodes and with preserving dual graph then we have $dimDef_G(\...
5
votes
1answer
196 views
Bernstein-Sato polynomial
Let $f$ be a polynomial. It is well-known that there exits a polynomial $b_f(s)$, such that $P\cdot f^{s+1}=b_f(s)f^s$ for some differential operator $P$. The polynomial $b_f(s)$ has been studied very ...
7
votes
1answer
119 views
Lie algebra preserving ideal of functions
Let $G$ be an algebraic group acting on an affine variety $X=\operatorname{Spec}A$ (all over $\mathbb{C}$). This gives an action of $G$ on the $\mathbb{C}$-algebra $A$, and an action of the Lie ...
6
votes
1answer
223 views
Inequality number of facets simplicial complex
In a recent preprint, Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies a version of ...
4
votes
0answers
97 views
Bertini-type theorem for strict transform
Let $(X,o)$ be an isolated, normal singularity of dimension at least $3$. Let $\pi: \widetilde{X} \to X$ be a resolution of singularity of $X$. Is it true that for a general hypersurface $H \subset X$ ...
3
votes
0answers
170 views
Singularities of rational quartic surfaces
Let $X\subset \mathbb{P}^3$ be an irreducible quartic surface, defined over an algebraically closed field $k$. Suppose that $X$ is rational (i.e. birational to $\mathbb{P}^2$). Is is true that $X$ has ...
2
votes
1answer
88 views
Definition of model functions and their density in $C^0(X^\text{an})$
I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10....
1
vote
0answers
94 views
Moduli interpretation of the integral anticanonical tower
This question is related to my reading of On torsion in the cohomology of locally symmetric varieties by Scholze.
In chapter $3$, using the theory of canonical subgroup, he produces Frobenius maps ...
1
vote
0answers
72 views
Notions of connected components in a finite family fibration
Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...
6
votes
1answer
439 views
Quaternion algebra actions on $\ell$-adic cohomology
Let $E$ be a supersingular elliptic curve over $\mathbf{F}_p$, and $H$ its endomorphism algebra $\text{End}(E)\otimes_{\mathbf{Z}}\mathbf{Q}$, a quaternion algebra (non split at $p$ and $\infty$).
...
0
votes
1answer
156 views
Counting the number of poles for rational functions in a coordinate ring of a curve
I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...
1
vote
0answers
61 views
Relation between lifts of simple roots and lifts of idempotents (Henselian property)
Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
19
votes
1answer
570 views
What are the equations for $SL_3/SL_2$?
Consider $SL_2$ embedded into $SL_3$ as upper left block matrices. The quotient $SL_3/SL_2$ is an affine variety, as is any quotient of reductive groups. How does one describe $SL_3/SL_2$? What are ...
8
votes
1answer
230 views
Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?
Let $k$ be a field of characteristic $0$. Let us denote by $\mathbf{1}_{k}$ the sphere spectrum. Let $MGL$ be the algebraic cobordism spectrum.
We have the following diagram
$$H\mathbb{Z}\...
5
votes
0answers
161 views
Complex manifolds as algebro-geometric objects
A result of Artin states that analytification of proper algebraic spaces over
$\mathbb{C}$ defines an an equivalence of the category of proper algebraic spaces with the category of Moishezon spaces. ...
2
votes
1answer
94 views
Naive compactification of $\mathbb{C}^*$-fibrations
Let $\pi:X \to Y$ be a $\mathbb{C}^*$-fibration between complex manifolds in the sense that there exists a fixed integer $a$ such that for every $y \in Y$, $\pi^{-1}(y)=(\mathbb{C}^*)^a$. Suppose ...
1
vote
1answer
170 views
When are two elementary transforms isomorphic?
Let $C$ be a smooth projective curve and $X=\mathbb{P}_C(E)$ be a ruled surface over $C$.
Let $x_1,\ x_2\in X$ be closed points and define $X_1,\ X_2$ to be elementary transforms of $X$ at $x_1,\ x_2$,...
3
votes
0answers
54 views
Fuchsian groups of singly branched covers
Let $X/\mathbb{C}$ be an algebraic curve with genus $g \geq 2$. Then by the uniformization theorem, with $X(\mathbb{C})$ viewed as a Riemann surface, it can be realized as the quotient $\mathbb{H}/\...
7
votes
1answer
332 views
Are higher etale homotopy groups topological groups in a natural way?
Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. ...
4
votes
0answers
107 views
Deformation of pairs (X,D) isotrovial along D
I have a log pair $(X,D)$ which is purely log terminal and $D$ is a projective $\mathbb{Q}$-Cartier divisor ($X$ may not be projective). Moreover, $D$ is a variety of Fano type. Is there a space of ...
1
vote
0answers
98 views
Universal elliptic curve over anticanonical tower
While I'm reading Scholze's paper "On torsion in the cohomology of locally symmetric varieties" he constructs the anticanonical tower passing through the construction of an integral model $X_{\infty}$ ...
3
votes
0answers
70 views
Restricting divisors to closed fiber of relative curve over henselian DVR
Setup:
$k$ is an algebraically closed field.
$\mathcal{O} = k\{t\}$ is the henselization of $k[t]_{(t)}$.
$V \rightarrow \text{Spec}(\mathcal{O})$ is proper and has a section.
$V$ is irreducible, ...
1
vote
0answers
94 views
Quasi-compactness of irreducible separated scheme locally of finite type
Is an irreducible separated scheme locally of finite type necessarily quasi-compact?
1
vote
0answers
80 views
singular $m$-canonical divisors
[remark for v2] I began by considering curves in v1. I am convinced that the answer is positive. Thanks to Jason Starr and abx.
Let $X$ be a complex projective variety.
Let $K_X$ be its canonical ...
1
vote
1answer
237 views
Finiteness of surjective etale morphisms
Is every surjective etale morphism from a connected separated scheme to $A^n_{\mathbb{C}}$ of finite type? Is it finite? We use Stacks project's definitions.
EDIT: From Jason Starr's answer, we ...
2
votes
1answer
79 views
Gysin morphism of blow up
Let $X$ be a smooth, projective variety and $i:Y \hookrightarrow X$ a smooth divisor. Let $Z \subset X$ be a proper, closed subvariety disjoint from $Y$. Let $\pi:\widetilde{X} \to X$ be the blow-up ...
