# Questions tagged [ag.algebraic-geometry]

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

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

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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 ...

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$\DeclareMathOperator\Ring{Ring}\DeclareMathOperator\Aff{Aff}\DeclareMathOperator\op{op}\DeclareMathOperator\Sch{Sch}\DeclareMathOperator\Zar{Zar}$The category of schemes sits, fully faithfully, in ...

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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}...

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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}^...

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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 ...

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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 ...

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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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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 ...

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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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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-...

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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\...

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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 $\...

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

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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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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 ...

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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 $\...

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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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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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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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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 ...

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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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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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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 $\...

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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 (...

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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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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 ...

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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 $\...

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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 $...

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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 $\...

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$\DeclareMathOperator\Sh{Sh}$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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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 ...

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$\def\CC{\mathbb{C}}\def\RR{\mathbb{R}}$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 ...

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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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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 $...

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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 ...

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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 ...

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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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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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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 $f_{Y_1}$...

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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 ...

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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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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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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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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 ...

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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 ...