Questions tagged [ag.algebraic-geometry]

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

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Vector subbundles of a given one in $\mathbb{CP}^1$

I apologize if this question is not suited for MathOverflow. This has been crossposted in MathStackExchange here and it is related to some open questions on that site that remain unsolved. I would ...
6 votes
0 answers
117 views

Epimorphisms and quotients in Sch versus $\mathrm{Sh}(\mathrm{Ring}^{\mathrm{op}},\mathrm{Zar})$

$\DeclareMathOperator\Ring{Ring}\DeclareMathOperator\Aff{Aff}\DeclareMathOperator\op{op}\DeclareMathOperator\Sch{Sch}\DeclareMathOperator\Zar{Zar}$The category of schemes sits, fully faithfully, in ...
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6 votes
1 answer
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On actions of finite groups on adic spaces

Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
1 vote
1 answer
153 views

Short exact sequence of equivariant line bundles on $\mathbb P^1$

I have a two-dimensional vector space ${\mathbb C}^2$ with basis $e_m, f_1$ and action of ${\mathbb C}^*$ by $t \cdot e_m = t^m e_m$ and $t \cdot f_1 = f_1$ and I have the projective line ${\mathbb P}^...
5 votes
0 answers
184 views

Do algebraic tori have no $H^1$?

If $G$ is an algebraic group over a field $K$, we can consider the Galois (or flat) cohomology $H^1(K, G)$. If $G = \mathbb{G}_a$ or $\mathbb{G}_m$, it is well known that $H^1(K, G) = 0$ (the latter ...
3 votes
0 answers
116 views

Can you determine the least degree of a morphism between algebraic curves?

I have several questions regarding the degrees of morphisms between algebraic curves. If we have algebraic curves $X$ and $Y$ defined over some perfect field $k$, can we determine the least degree of ...
2 votes
0 answers
100 views

Homotopy invariant Bloch-Ogus cohomologies with a vanishing property

I am looking for examples (in any characteristic) of homotopy invariant Bloch-Ogus cohomology theories given by Zariski sheaves $\Gamma(n)$, such that $\Gamma(0) = \mathbb{Z}$ is the constant sheaf. ...
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2 votes
1 answer
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Irreducible components of a general singular fiber correspond to irreducible components of the hypersurface consisting of singular fibers

I already asked this on math.SE, but didn't receive any response. The following question arose when studying Hwang and Oguiso's Characteristic foliation on the discriminant hypersurface of a ...
3 votes
0 answers
121 views

On the notion of rational singualrities over positive characteristic

Let $k$ be an algebraically closed field and $X$ be a normal variety. According to [Kol13], the notion of rational singularities may be defined as follows: We say $X$ has rational singularities if ...
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6 votes
1 answer
271 views

How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?

In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-...
16 votes
2 answers
857 views

Strengthening Ax-Grothendieck

The question was cross-posted from Math.SE: https://math.stackexchange.com/questions/4566017/strengthening-ax-grothendieck The question is simple. The Ax-Grothendieck theorem says a polynomial map $p\...
3 votes
0 answers
72 views

Explicit example of wall-crossing for sheaves

I would like to see an explicit example of a coherent sheaf $\mathcal{E}$ on a projective complex threefold $X$ which crosses the wall of stability. That is, I would like some $1$-parameter family $\...
0 votes
1 answer
122 views

Does $\mathcal{F}|_{x\times Y}\cong\mathcal{G}|_{x\times Y}\Rightarrow q^*\mathcal{M}\otimes\mathcal{F}\cong q^*\mathcal{N}\otimes\mathcal{G}$

Consider a product of projective varieties $X\times Y$ and two cohernet sheaves $\mathcal{F},\mathcal{G}$ such that $$\mathcal{F}|_{x\times Y}\cong\mathcal{G}|_{x\times Y}$$ for any $x\in X$. Thanks ...
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1 vote
0 answers
58 views

Abelian coverings as pull-backs of isogenies

Fix an algebraically closed field $k$ and a smooth projective (geometrically) integral genus $\geq 2$ curve $C$ over $k$. Denote by $J_C$ the Jacobian variety of $C$. Chapter VI, Proposition 11 of ...
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1 answer
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Calculate blowup of a pencil of cubics "by hand"

I have one more question about the Example (I.5.1) on page 7 from Rick Miranda's the basic theory of elliptic surfaces: Let $C_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C_2$ be any other ...
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13 votes
2 answers
325 views

Tensor product of finite type UFD algebras over an algebraically closed field is again UFD?

Let $K$ be an algebraically closed field, $A$ and $B$ two finite type $K$-algebras which are assumed to be UFD. Is $A \otimes_K B$ again a UFD? This question has been already asked here and here, but ...
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1 vote
0 answers
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Cohomology of a curve and its Jacobian over an algebraic closure of a number field

In this MathOverflow post, the smooth projective curve $C$ was defined over $\mathbb{C}$ and we have an isomorphism of de Rham cohomology groups $$H^1(C, \mathbb{C}) \cong H^1(J_C, \mathbb{C}),$$ ...
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2 votes
0 answers
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Is $\mathbb{Z}_{\mathrm{tr}}(X)$ a cdh sheaf?

Suppose $X\in \mathrm{Sm}/k$. Is the sheaf with transfers $\mathbb{Z}_{\mathrm{tr}}(X)$ a cdh sheaf? Its sections are finite correspondences.
2 votes
1 answer
156 views

What is the space of maps between superplane $\mathbb{R}^{0|2}$ and a smooth manifold $M$?

Within the algebrogeometric approach to supergeometry, a supermanifold of dimension $m|n$ is an ordinary $m$ dimensional smooth manifold $M$ and a sheaf of supercommutative super algebras $\mathbf{C}^...
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A coherent sheaf is generated by global sections on fibres

Suppose $T$ is a $k$-scheme of finite type,and $\mathcal K$ is a $T$-flat coherent sheaf on $P_{T}^{n}$ . If for any point $t \in T$, $\mathcal K|_{P_{k(t)}^{n}}$ is generated by its global ...
0 votes
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132 views

How to reduct the case $\mathcal Quot_{Y/k}(\mathcal F ,P)$ to $\mathcal Quot_{P^{n}/k}(i_{*}\mathcal F ,P)$

Let $X$ be a projective scheme over $k$.And let $i$: $X \rightarrow P^{n}$ be a closed embedding.Then for any coherent sheaf $\mathcal F$ over $X$ and any polynomial $P \in Q[Z]$.I can prove $\...
6 votes
2 answers
417 views

When Atiyah class and Chern class coincide?

Let $X$ be a compact complex manifold, $L$ be a holomorphic line bundle on $X$, then the exponential exact sequence $0\to \mathbb Z\hookrightarrow \mathcal O\to \mathcal O^*\to 0$ induces the map $c:H^...
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3 votes
0 answers
102 views

Resolving the "wild" singularities of $\mathbb A^n/C_n$

Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
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1 vote
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112 views

What is the lattice of the field $\mathbb Q_p(\sqrt[p^5-1]{p^2})$?

Let $p$ be odd prime and $\mathbb Q_p$ be the $p$-adic field. Consider the field extension $K=\mathbb Q_p(\sqrt[(p^5-1)]{p^2})$ of $\mathbb Q_p$ of degree $\frac{p^5-1}{2}$. My question: I want see ...
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5 votes
0 answers
309 views

Perfect algebraic spaces on a paper of Xinwen Zhu

I have problem reading Xinwen Zhu's paper Affine Grassmannians and the geometric Satake in mixed characteristic about perfect algebraic spaces in Section A.1. Let $k$ be a perfect field of ...
1 vote
0 answers
68 views

Lower bound of degree of ruled surface in $\mathbb P^n$

I have a question of Complex Algebraic surface in Beauville. Let $S\subset\mathbb{P}^n$ be a (birationally) ruled surface of degree $d$ lying in no hyperplane. Show that $d\geq 2 n-2$ if $S$ is not ...
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2 votes
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110 views

Any kind of duality between differentials and Tate modules?

Let $X$ be a curve over some algebraically closed field $k$ and let $J$ be its Jacobian. I have read that one should think of the Tate module $T_lJ$ as being the first homology group of $X$ with ...
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1 vote
0 answers
80 views

An elliptic threefold and the Mordell–Weil lattices of its reductions

Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\...
1 vote
1 answer
108 views

Picard group of an elliptic fibration is generated by multisection and fibres

Assume that $C$ is a projective curve and $X$ is an elliptic fibration over $C$. What is the picard group of $X$? can we say something about it? I think it should be generated by multisections and (...
user avatar
1 vote
1 answer
112 views

Projective embedding of a compact complex surface

Let $M$ be a compact complex surface which admits a holomorphic line bundle $L$ with $c_1^2(L)>0$. Can we prove that $M$ is projective?
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3 votes
1 answer
116 views

Algebraic representations and vector bundles

This might seem like a silly question considering my relatively elementary knowledge of representation theory. The question is regarding Eugen Hellman 's paper titled "On the derived category of ...
25 votes
4 answers
3k views

Why do we care about the eigenvalues of the Frobenius map?

The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
7 votes
1 answer
181 views

Reference request. Finiteness of the Selmer group

Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $...
4 votes
1 answer
247 views

Can the functor of the points of a scheme be characterized by its values ​on subcategories of the affine schemes?

A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site. Suppose $\...
5 votes
0 answers
140 views

Do quasi-excellent rings have a good constructive definition?

$\DeclareMathOperator\Sh{Sh}$Informally, a quasi-excellent ring is a Noetherian ring with a few technical extra regularity conditions that make it turns out to be the largest class of rings that allow ...
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2 votes
0 answers
57 views

What should I call a log scheme with free reduced monoids?

This is a terminology question about a class of log varieties. Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a ...
9 votes
2 answers
373 views

A category of spaces in which $\mathbb{R}_{>0}$ is not isomorphic to $\mathbb{R}$

$\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}$I am writing up notes on totally positivity in flag varieties. I often have the following situation: I have a complex variety $X$ defined over $\RR$ and a real ...
9 votes
1 answer
343 views

$H^{1,1}(X)$ in positive characteristic

Let $X$ be a smooth projective variety over an algebraically closed field $k$. When $k= \mathbb C$, it is known that $h^{1,1}(X)=h^1(X,\Omega_X) > 0$. Is this true also when $\rm char(k)=p > 0$? ...
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1 vote
1 answer
91 views

Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve

We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...
2 votes
0 answers
117 views

On the generalization of a Cech-to-sheaf type spectral sequence

Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
5 votes
0 answers
145 views

On the elementary proof of Dirichlet theorem on arithmetic progressions

In [Cassels, JWS, Rational quadratic forms, p. 333], the autor says: "In fact the elementary proof of Dirichlet's theorem [Selberg (1949)] makes essential use of the existence of genera". In ...
1 vote
0 answers
70 views

Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$

Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...
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1 vote
1 answer
126 views

Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces)

Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument that I do not understand yet. Let $C_1$ be a smooth ...
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0 votes
0 answers
79 views

Extending a line bundle from a subvariety to the union

If I have a smooth projective variety $X$, where $Y_1$ and $Y_2$ are subvarieties, where they have same codimension in $X$. If I have a line bundle $L$ on $Y_1$, and non-zero rational section $f_{Y_1}$...
3 votes
1 answer
211 views

Groebner basis with parameters

I need to compute a Groebner basis of a polynomial system with parameters. The only recent results I found is Groebner cover: https://www.sciencedirect.com/science/article/pii/S0747717110000970 Are ...
0 votes
0 answers
95 views

What is a rational cross-section?

I'm reading "Topics in Galois Theory" from Serre, Chapter 3. I'm trying to understand the equivalent definition of thin set but can't find the definition of a "rational cross-section&...
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5 votes
1 answer
161 views

What is the surface area of the finite part of the Cayley nodal cubic surface?

The Cayley nodal surface is defined by the equation $x^2+y^2+z^2-2xyz=1$. The finite part of the surface is the tetrahedral part bounded by the 4 nodes $(1,1,1)$, $(1,-1,-1)$, $(-1,1,-1)$, $(-1,-1,1)$....
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15 votes
1 answer
956 views

How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?

In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
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3 votes
0 answers
219 views

What are some concrete applications of Grothendieck's six operations?

In Gallauer's An introduction to six-functor formalisms I read: Indeed, the language and theory of six-functor formalisms permeates much of modern algebraic geometry and beyond, and has spawned ...
1 vote
0 answers
103 views

The geography of models of Fano varieties

This question aims to compute ${\rm Vol}(-K_X-tD)$ where $X$ is a $\mathbb{Q}$-factorial Fano variety of dimension $n$ and $D$ is a nonzero effective divisor on $X$. This volume is positive when $0\le ...
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