Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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6 votes
1 answer
214 views

Rational points on regular curves over global fields

Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy: If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...
0 votes
0 answers
44 views

0th and 1st cohomology of sheaf of differentials

Let $M$ be the moduli space of pricipal $G-$bundles, where $G$ is a complex semisimple Lie group on a smooth projective curve over the field of complex number of genus $\ge 2$. Question: Are $h^1(M, \...
0 votes
0 answers
61 views

Cohomology of sheaf of differentials on singular Fano variety

It is known that if $X$ is a smooth Fano Manifold of Picard number one, then $H^0(X, \Omega_X^1) = 0$ and $H^1(X, \Omega_X^1) \simeq \mathbb{C}$. Question: Are the same results hold true if $X$ is a ...
9 votes
0 answers
336 views

Finiteness for motivic local systems

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$. Say a complex local system $\mathbb{V}$ on $X$ is motivic if there exists a dense Zariski-open subset $U\subset X$, and a smooth proper ...
4 votes
2 answers
384 views

Basic question on projective bundles

Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
0 votes
0 answers
16 views

A difficult problem about converting parametric equations to cartesian [migrated]

this is not a homework question and I consider this problem quite difficult and confusing. I tried hard to solve it for 2 days, sure I found solutions, but they are not the same that the one provided ...
1 vote
0 answers
87 views

K-theory of l-adic sheaves of a curve

I am trying to understand if there is a good notion of a $K$-theory attached to the etale topology on a nice scheme $X$ (say smooth projective goem connected curve over a finite field is enough for me)...
3 votes
0 answers
51 views

Auslander-Reiten translate of toric vector bundles

By the result of A. Klyachko, there is categorical equivalence of toric vector bundles i.e. t-equivariant vector bundles on toric varieties and filtered vector spaces. For instance, on a toral surface,...
0 votes
0 answers
60 views
+300

explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety

Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and ...
11 votes
1 answer
678 views

Cohomological bounds for scalar curvature of an extremal Kähler metric

There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...
1 vote
0 answers
54 views

Example of nontrivial families of isolated singularities with constant Milnor number

In Lê-Ramanujam's paper The invariance of Milnor’s number implies the invariance of the topological type, they prove what the title says for families with isolated singularities and constant Milnor ...
7 votes
2 answers
1k views

Coherent sheaves and holomorphic vector bundles

For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its category of coherent sheaves is. If I understand correctly, then the sheaf of holomorphic ...
11 votes
6 answers
2k views

How to get explicit unramified covers of an elliptic curve?

Say we begin with an explicit elliptic curve over $\mathbb{C}$, say: $y^2=x(x-1)(x-2)$. According to abstract reasoning this elliptic curve has an (several, in fact) unramified cover with group $C_n$. ...
9 votes
1 answer
701 views

Are there non-projective normal surfaces which are rational?

Every non-singular complete surface is projective. On the other hand, there are non-projective complete surfaces (see e.g. Excercise II.7.13 of Hartshorne) - and there are such examples where the ...
2 votes
0 answers
64 views

Unramified section associated to a rational point

This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale ...
3 votes
2 answers
310 views

Functor between categories of motives

$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Motives{Motives}$Let us assume for the moment that we have a "nice" category of motives, that is for fields $k$ we have a contravariant ...
3 votes
0 answers
124 views

Linear deformations of a morphism between stacks

Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$? In ...
1 vote
1 answer
176 views

Counterexample to purity of Brauer group for curves

The purity of Brauer group states that for a smooth (quasi-)projective variety $X$ over a field $k$, removing a closed subscheme $Z \subset X$ of codimension at least $2$ ensures that the restriction ...
5 votes
2 answers
290 views

Modular parametrization abelian varieties

Let $N$ be a positive integer, and let $f$ be a newform for $S_2(\Gamma_0(N))$. Then by Shimura's construction, the variety $J_0(N)$ has a quotient $A_f$ which is an abelian variety attaced to $f$. $$\...
6 votes
2 answers
1k views

Cohomologie Etale

Is there an English translation available for Deligne's Cohomologie Etale (Arcata) that is now part of the SGA 4 1/2 ?? Atleast for the first two sections - Grothendieck Topologies and Etale Topology.
64 votes
2 answers
8k views

Who is the "young student" André Weil is referring to in his letter from the prison?

I am reading a nice booklet (in Italian) containing the exchange of letters that André and Simone Weil had in 1940, when André was in Rouen prison for having refused to accomplish his military duties. ...
0 votes
0 answers
86 views

Morphism defined by sections of a globally generated vector bundle

Let $X$ be a smooth projective variety of dimension $n$ over the field of complex numbers and $E$ be globally generated vector bundle of rank $n$. Choose a general $n$-linearly independent sections ...
3 votes
1 answer
142 views

Perverse tilting sheaves

In the article titled Tilting Exercises (See http://arXiv.org/abs/math/0301098v3) the authors define a notion of tilting perverse sheaves on an algebraic vareity $X$ with respect to stratification $\...
1 vote
0 answers
41 views

Exponential sums of fully singular hypersurfaces

Let $p \in \mathbb{Z}$ be a prime and let $\mathbb{F}_p$ be the field with $p$ elements. Let $\chi: \mathbb{F}_p \to \mathbb{C}$ be an additive character, and let $f \in \mathbb{F}_p [x_1, \dots, x_n]$...
2 votes
0 answers
72 views

Reconciling two notions of finite descent obstructions

Let $k$ be a number field and $X$ a smooth geometrically connected variety over $k$. We denote by $H(k,X)$ the set of sections $G_k \rightarrow \pi_1(X)$, where $G_k$ is the absolute Galois group of $...
7 votes
1 answer
413 views

Intuition for Luna's Étale Slice Theorem

I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$. Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group ...
2 votes
1 answer
502 views

A "boundary map" for the algebraic equivalence relation of cycles

In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want. Let $X$ be ...
3 votes
2 answers
155 views

How do we define the type of a singularity on a cubic surface?

Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the ...
1 vote
1 answer
103 views

Embedding singular divisors into smooth varieties

Let $X=X_{d_1}\cap\cdots \cap X_{d_m}\subset \mathbb{P}^n$ be a smooth complete intersection of smooth hypersurfaces $X_{d_i}$ of degree $d_i$ with $1<d_1\leq ...\leq d_m$ and $m>1$. For ...
3 votes
1 answer
113 views

Normalisers and stabilisers in classical groups $\operatorname{PGL}_{4}$

In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{...
1 vote
0 answers
82 views

Isomorphism class of an $\operatorname{Aut}(G) $-torsor

Let $X$ be a smooth projective geometrically connected curve over a field $k$ and $x\in X$ a geometric point. Let $G$ be connected reductive group over a $k$ and let $\underline{G} = G\times X$ étale-...
16 votes
3 answers
2k views

Why do Grothendieck topologies used in algebraic geometry typically involve finiteness conditions?

There are many Grothendieck topologies used in algebraic geometry, with complex interrelations. Generally in one of these topologies, a cover of schemes is a family of maps which is jointly surjective ...
0 votes
0 answers
78 views

Is co-tangent bundle of a homogeneous spaces are completely integrable system?

Let $M= T^*(G/P)$ where $G$ is an algebraic group and $P$, a parabolic subgroup. Let $\mathbb{g}$ be its Lie algebra. Then there is moment map $\mu: M \to \mathbb{g}^*$, where $\mathbb{g}^*$ is the ...
1 vote
0 answers
87 views

branched cover over a quadric surface

We assume $k=\mathbb{C}$. Consider $X=\mathbb{P}^1\times\mathbb{P}^1$ and a smooth divisor $D=V(s)$ for some $s\in H^0(X,\mathcal{O}(d,d))$ for $d>1$. Let $\pi:S\rightarrow X$ be the $d$-th ...
-1 votes
0 answers
100 views

Question on a Lagrangian fibration

Let $(M, \omega)$ be a symplectic manifold. A Lagrangian fibration is a surjective map $\pi :(M, \omega) \to B$ whose regular fibres are Lagrangian submanifolds of $(M, \omega)$, which implies that $\...
0 votes
1 answer
189 views

Is there a non-singular cubic surface that has a point where four lines intersect?

Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt ...
6 votes
1 answer
267 views

Has anyone attempted to generalize the notion of a higher differential of $ A $ and the sheaf of differentials $ \Omega_{A/k} $?

Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-...
3 votes
0 answers
135 views

Examples of completely integrable system

Let $X$ be a Fano manifold (different from $\mathbb{P}^n$) of dimension $n$ and Picard rank one. If its cotangent bundle $T^*X$ admits $n$ independent Poisson commuting regular functions then $T^*X$ ...
4 votes
0 answers
279 views

What are some fundamental papers in derived algebraic geometry for a beginner?

If you could recommend a few papers for someone entering derived algebraic geometry outside of the classical category theory, algebraic geometry, and algebraic topology sequences? What would they be?
2 votes
0 answers
60 views

Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$

The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is $$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$ ...
0 votes
0 answers
70 views

Coherent sheaves with the same chern class as a line

Let $X=\mathbb{P}^n$ over $\mathbb{C}$ for $n>1$, we know that there exists a line on $\mathbb{P}^n$ say $L$. Could we have a coherent sheaf $\mathcal{F}$ other than $\mathcal{O}_L$ (and that for ...
0 votes
0 answers
89 views

Smoothness of the moduli space of principal G-bundles

Let $X$ be a smooth projective curve over the complex numbers. Question: When the moduli space of principal $G-$bundles over $X$ is smooth Fano of Picard rank one ?
1 vote
0 answers
87 views

Some questions about inclusions of Brauer groups/sets

I have a feeling this question will be left hanging on StackExchange, so I've decided to just post it here, let me know if it's not suitable. First of all, I'm trying to show that given an immersion ...
1 vote
0 answers
87 views

A question about Galois group action on sheaves, descent theory and $\mu$-semistable sheaves

Let $f : Y \to X$ be a finite morphism of degree $d$ of normal projective varieties over $k$ of dimension $n$. In Lehn and Huybrechts' book "The Geometry of Moduli Space of Sheaves", there ...
0 votes
1 answer
163 views

Compute the cone of $\mathcal{O}_X\rightarrow\mathcal{O}_X(d-n-2)[n]$

Consider a smooth hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $d$ over a nice field (such as $\mathbb{C}$), we know that the cone $C(\operatorname{id}:\mathcal{O}_X\rightarrow\mathcal{O}_X)=0$. ...
40 votes
2 answers
7k views

Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture

Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635 In that preprint, Kirti Joshi claims that he agrees with Scholze and ...
10 votes
4 answers
2k views

Question on Kähler/ample cone, cone of curves....

Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$). Let's $NE(X)$ be the cone of effective 1-...
2 votes
0 answers
158 views

Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?

A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...
2 votes
0 answers
63 views

Steenbrink spectral sequence and modifications of the central fibre

If $f: X \to S$ is a proper map from a complex manifold to a disc, $Y=f^{-1}(0)$ is a divisor with strictly normal crossings and the action of monodromy on $X_t=f^{-1}(t)$ for some (hence any) $t \neq ...
1 vote
0 answers
67 views

Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves

Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...

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