# Questions tagged [ag.algebraic-geometry]

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

16,656
questions

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11 views

### 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 ...

**1**

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**1**answer

69 views

### 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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32 views

### 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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138 views

### 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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68 views

### 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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64 views

### 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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43 views

### 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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137 views

### 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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75 views

### 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 $z_1$, ....

**14**

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**1**answer

367 views

### 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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117 views

### 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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**1**answer

157 views

### An explicit negative solution to the Lüroth problem for non-algebraically closed fields

Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$.
If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \...

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93 views

### Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)

I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2
& Example 12.9.2):
Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point
$(0:...:1)...

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44 views

### 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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95 views

### 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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48 views

### 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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96 views

### 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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**1**answer

203 views

### 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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**1**answer

135 views

### 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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**1**answer

138 views

### 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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79 views

### 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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**1**answer

112 views

### 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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36 views

### 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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53 views

### 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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89 views

### 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 ...

**2**

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**1**answer

152 views

### Question about automorphism functor in Sernesi's “Deformations of algebraic schemes”

Let $X$ denote an algebraic scheme over $\operatorname{Spec} k$ such that its deformation functor $\operatorname{Def}_X$ has a semi-universal couple $(R,u)$, where $R$ is an Artinian $k$-algebra and $...

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97 views

### Log Calabi-Yau variety diffeomorphic to an algebraic torus

Let $U$ be a complex affine log Calabi-Yau variety, which I take to mean a smooth affine variety which admits a compactification in a smooth projective variety $X$ with an snc anti-canonical ...

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80 views

### Local freeness of $\pi_*F(r)$ from flatness of $F$

In 'Fundamental Algebraic Geometry' by Fantechi there is a lemma in section 5.3.2, page 119:
LEMMA 5.5 Let $S$ be a noetherian scheme and let $F$ be a coherent sheaf on $\mathbb{P}^n_S$. Suppose there ...

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68 views

### Toric ideals are generated by binomials. $V(x)$ gives affine $n-1$ space in affine $n$-space. $x$ is not a binomial, yet affine $n-1$ space is toric?

Proposition 1.1.11 of Cox-Little-Schenk's Toric Varieties states that an ideal $I \subseteq \mathbb{C}[x_1, \dots, x_n]$ is toric iff it is prime and generated by binomials. Setting $I = (x_1) \subset ...

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84 views

### 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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75 views

### 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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89 views

### 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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60 views

### 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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100 views

### 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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83 views

### 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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44 views

### 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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177 views

### Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar.
Define an $\mathrm{SL}(2)$-action on $\...

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169 views

+100

### A density criterion and a submersion map of a Hodge bundle

In Voisin's excellent book 《Hodge theory and complex algebraic geometry II》5.3.4 - a density criterion, there is a important theorem:
Let $X$ be a compact Kähler manifold, $\pi:\mathcal X \rightarrow ...

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**3**answers

830 views

### 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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104 views

### Scheme-theoretic image of the inverse image of a morphism of schemes

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 \...

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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(...

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**1**answer

205 views

### On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group

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 ...

**3**

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**1**answer

102 views

### Are “strongly finite dimensional” homotopy invariant sheaves with transfers (locally) constant?

Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...

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141 views

### Riemann hypothesis for the motivic zeta function

To repair the failure of rationality in general (as shown by Larsen and Lunts for products of two curves of genus > 1) of M. Kapranov's zeta function defined for a variety over a field $k$ and ...

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149 views

### Why doesn't the Manin obstruction work for quadratic forms?

The Manin obstruction explains why the Hasse principle doesn't work for non-quadratic forms. Let's write the notation first;
$V(\mathbb{Q})$ is variety for rational numbers.
$V(A_\mathbb{Q})$ is ...

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198 views

### Galois action on torsion in homotopy groups not induced by homotopy equivalences

Let $V$ be a simply connected smooth projective complex variety defined over the rationals. Then for any integer $n\geq 2$ the group $\pi_n(V)$ is finitely generated abelian so profinite completion ...

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58 views

### Question regarding Algebra problem [closed]

"T.L. and Quina plan to add a 6-inch thick layer of gravel to their driveway. The driveway is 9 yards long and 3 yards wide. What volume of gravel in cubic yards is required?"
My approach to ...

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162 views

### Some computational results and goals of stable motivic homotopy theory of schemes

I am trying to learn ($\mathbb{P}^1$-)stable motivic ($\mathbb{A}^1$-)homotopy theory of schemes from the Cisinski-Deglise book, Triangulated Categories of Mixed Motives. In order to keep myself going ...

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174 views

### A homotopy equivalence from a variety to itself that is not homotopic to a homeomorphism

Let $V$ be a simply connected smooth projective complex variety. Can there be a homotopy equivalence $V\to V$ that is not homotopic to a homeomorphism?

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324 views

### Complex conjugation inducing a trivial map on the fundamental group

Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by ...