# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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### Algebraically cutting out continuous functions from all functions $\mathbb{R}\to\mathbb{R}$

Denote by $S$ the set of closed points in $X=\mathrm{Spec}\:\mathbb{R}[x_\alpha]$ ($\alpha \in \mathbb{R}$) that have $\mathbb{R}$ as the residue field. There is an obvious bijection from $S$ to the ...
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### Density of integral points on affine cubic surfaces of a certain type

Let $Q(x,y,z)$ be a cubic polynomial with integer coefficients, such that the terms $x^3, y^3, z^3$ do not appear. That is, it is at most quadratic in each of the variables $x,y,z$. Is there a general ...
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### Understanding a step in proof of how the localization of an additive category by a subclass of morphism satisfies Ore is also additive

I started to study localization in additive and triangulated categories via a subclass of morphism which satisfies the Ore conditions by my own. Right now, Im studying how for an additive category $C$...
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### Varieties isomorphic $\mathrm{mod}\:p$ are diffeomorphic

If two smooth proper varieties over $\mathbb{Q}$ have isomorphic smooth reductions modulo some prime (for some choice of integral models) are they diffeomorphic after tensoring with $\mathbb{C}$?
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### When every localization of the polynomial ring over a ring has finitely many idempotents

Let $R$ be a commutative ring such that every localization ring $R_r$ has finitely many idempotents for each non nilpotent element $r\in R$. Why dose every localization ring $R[x]_{f(x)}$ have ...
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### Slope stability using the Riemann-Roch theorem

I am studying about the (Stability on a curve). Suppose $C$ is a smooth curve of genus g. The Riemann-Roch theorem asserts that if $E$ is a coherent sheaf on $C$ then the Euler characteristic of $E$ ...
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### Special fiber of a reflexive sheaf over DVR

Let $f:X \to \mbox{Spec}(R)$ be a flat, projective morphism with $R$ a discrete valuation ring and the special and generic fibers of $f$ are normal and integral. I am looking for examples of rank $1$, ...
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### Homotopy type of the affine Grassmannian and of the Beilinson-Drinfeld Grassmannian

The affine Grassmannian of a complex reductive group $G$ (for simplicity one can assume $G=GL_n$) admits the structure of a complex topological space. More precisely, the functor $$X\mapsto |X^{an}|$$ ...
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### Questions related to compact complex curves, symmetric products and linear independence

Let $X$ be a compact complex curve and let $L$ be a very ample line bundle over $X$. Denote by $C_n( X )$ the configuration space of $n$ (ordered) distinct points on $X$. Given distinct points $z_1$, ....
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### Non-algebraic holomorphic maps between algebraic curves

Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic ...
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### Holomorphic map from a punctured affine line to $M_g$ with Zariski-dense image

Denote by $M_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. Is there a holomorphic map $U\to M_g$ with Zariski-dense image where $U$ is the affine line with ...
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### Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. ...
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### Kan liftings and projective varieties

Regard the following two bicategories: $\operatorname{dg-\mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$-$D$-bimodules. Composition ...
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### Is the cohomology of Hilbert modular surfaces spanned by special cycles?

We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
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### An analogue of Noether's Problem for non-rational varieties

For the sake of simplicity, let $\mathsf{k}$ be algebraically closed and of zero characteristic. Varieties are irreducible. The (linear) Noether's Problem (which goes back to the early 1910's in ...
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### Can Hodge symmetry fail if there is a lift to $W_2$ and the crystalline cohomology is torsion-free?

Let $f:X\to \mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism with $p>\mathrm{dim}\:X$. Assume that $H^i_{\mathrm{crys}}(X/\mathbb{Z}_p)$ is torsion-free for all $i\geq 0$ and that there is ...
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### Is there an algorithm on an infinite time Turing machine to compute a dense subset of $M_g$ for large $g$?

Denote by $M_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. If $g$ is a random large integer (e.g. $g=100$) does there exist an algorithm on an infinite time ...
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### Can $h^{1, 0}$ and $h^{1, 1}$ jump for smooth projective surfaces over $\mathbb{Z}[1/N]$?

Let $N$ be a positive integer. Let $f:X\to S=\mathrm{Spec}\:\mathbb{Z}[1/N]$ be a smooth projective morphism of relative dimension 2 such that $R^1f_*\mathcal{O}_X$ and $R^2f_*\mathcal{O}_X$ are both ...
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### A normal proper model of an abelian variety with geometrically integral special fiber smooth at the reduction of the origin

Let $A$ be an abelian variety over $\mathbb{Q}_p$. Does there exist a proper flat morphism $X\to \mathrm{Spec}\:\mathbb{Z}_p$ such that the generic fiber is isomorphic to $A$, the special fiber is ...
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### Mean square estimate for the Kloosterman sums

For $m,n\in \mathbb{N}$, denote the Kloosterman sum $$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$ denotes the multiplicative inverse of $a\bmod c$. Does ...
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### Find Angle in Triangle [closed]

In triangle ABC, angle a = 56 degrees and angle B = 50 degrees. The altitude from B to AC is extended until it intersects the line through A that is parallel to BC; that intersection is called point K....
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### Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix

Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$. My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
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### Variation of Euler characteristic when the sheaf is not flat

Let $f:X \to Y$ be a flat, projective morphism with $Y$ integral and every fiber of $f$ normal and integral. Let $F$ be a torsion-free, coherent sheaf on $X$ (not necessarily flat over $Y$). Then, is ...
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### Holomorphic tubular neighborhood of divisors at infinity

For the discussion of holomorphic tubular neighborhoods and some criteria for their existence see this question. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. Hironaka tells us that ...
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### Counting quadratic curves in $\mathbb P^1 \times \mathbb P^1$ passing through seven points in general position

Let $p_1,\dots,p_7 \in \mathbb P^1 \times \mathbb P^1$ be 7 points in general position. What is the number of maps $F=(F_0,F_1):\mathbb P^1 \to \mathbb P^1 \times \mathbb P^1$ modulo domain ...
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### Pseudoreflection groups in affine varieties

Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result: (C-S-T): Let $G$ be a ...
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### When the sum of two ideals is indecomposable

I am looking for a commutative ring $R$ and two ideals $I$ and $J$ of $R$ and two different maximal ideals $m_1$ and $m_2$ of $R$ such that $ann(I)=m_1$ and $ann(J)=m_2$ and $I+J$ is an ...
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### Pushforward of the structure sheaf for smooth morphisms

Let $\pi:X\to S=\mathbb{P}^1_{\mathbb{Z}}$ be a smooth morphism. Is there a smooth separated morphism $\pi':X'\to S$ such that $\pi_*\mathcal{O}_X\approx \pi'_*\mathcal{O}_{X'}$ as $\mathcal{O}_S$-...
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### Introductory text for representations of compact/profinite groups

I am looking for a text on representation theory of topological (at least profinite) groups over fields (allowing the non-algebraically closed case). Reasonably selfcontained (or assuming a standard ...
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### Uniform position for multiple components

(Modified from https://math.stackexchange.com/questions/3730261/uniform-position-theorem-for-reducible-varieties/3730457#3730457) The uniform position theorem states (roughly) that a general ...
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### Are “large enough” finite etale covers arithmetic?

Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...
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Let $f:X \to Y$ be a finite, surjective morphisms between noetherian, integral varieties (over $\mathbb{C}$). I am looking for conditions on $f$ under which I can say that for any closed subscheme $Z \... 0answers 94 views ### pro-commutative group schemes When$k$is field, Demazure and Gabriel defined and worked with the category of commutative pro-algebraic groups over$k$. In their book, they proved that$Ext^n(\varprojlim G_i, H)= \varinjlim Ext^n(...
Background: I've seen two versions of the homotopy exact sequence for etale fundamental groups. One from Stacks: Stacks 0BTX: Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a ...