# Questions tagged [ag.algebraic-geometry]

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

16,812
questions

**8**

votes

**0**answers

209 views

### What is wrong with $A^{(2)}_{2n}$?

When dealing with affine Kac-Moody groups, especially geometrically (e.g. by examining their affine flag varieties or affine Grassmannians) I've been taught that time and time again, issues arise in ...

**1**

vote

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87 views

### Injective resolution of representable sheaves

Let $S$ be a scheme and consider the abelian category $\mathcal{A}$ of sheaves of abelian groups on the fppf site over $S$. It is known that $\mathcal{A}$ has enough injectives. My question is: given ...

**2**

votes

**1**answer

647 views

### Are cohomology functors sheaves?

Question is the following:
Is the functor $H^n_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$?
More generally, are cohomology functors sheaves in ...

**3**

votes

**0**answers

98 views

### Preserves naively $\mathbb{A}^{1}$-homotopic maps

I've been studying $\mathbb{A}^{1}$-homotopy recently and would like some guidance with the question below. Thank you so much.
Setup
Fix $k$ a field of characteristic zero. Let $Sm_{k}$ denote the ...

**4**

votes

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324 views

### Reference for a real algebraic geometry problem [migrated]

Disclaimer: I am not a mathematician by training.
I encountered the following problem in my research. Assume that I have $N$ real variables $x_1, x_2, \dots, x_N$. I am given $N$ homogeneous ...

**4**

votes

**1**answer

254 views

### Does big and nef imply projectivity?

Suppose that we have a compact Kaehler manifold $X$ with big and nef canonical class $c_1(K_{X})$, does it imply that $X$ is projective? By the base point free theorem, big and nef implies semi ample ...

**2**

votes

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242 views

### Rational homotopy type of a complex algebraic variety defined over $\mathbb{Q}$

Does there exist a simply connected smooth proper complex variety that is not rationally homotopy equivalent to a simply connected smooth proper complex variety defined over $\mathbb{Q}$?

**2**

votes

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145 views

### Algebraic groups as functors of points vs maximal points over algebraically closed field

So I only started learning about group schemes this summer, and I found two approaches. For the record I am interested in affine group schemes $G$ of finite type over a field $k$ (algebraic groups).
...

**3**

votes

**1**answer

217 views

### Analytic vs Zariski neighbourhood of a fibre

Let $f\colon X\to \mathbb P^1$ be a proper morphism of smooth complex algebraic varieties and let $p\in\mathbb P^1$. Are there a complex disk $\Delta\subseteq\mathbb P^1$ and a Zariski open subset $U\...

**4**

votes

**0**answers

100 views

### Hilbert scheme of real curves

Morally speaking, my question is whether every real smooth projective curve can be deformed in as many real directions as complex directions. Let me make the question precise.
Let $H$ be the Hilbert ...

**4**

votes

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113 views

### Quotient Jordan property

The Jordan property for finite subgroups of ${\rm GL}_n(\mathbb{C})$ says that there exists a constant $c(n)$ so that for any finite subgroup $G$ of ${\rm GL}_n(\mathbb{C})$ there is a normal abelian ...

**1**

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138 views

### Schlessinger criterion and finiteness of tangent space

Schlessinger's criterion allows us to study whether or not a functor $\mathcal{F}:C_{\Lambda}\rightarrow \text{Set}$ from the category of local Artin $\Lambda$-algebras to sets is representable. One ...

**2**

votes

**0**answers

99 views

### Computing whether a set of polynomials cuts out a projective variety

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a projective variety, i.e. whether the radical of the ideal $I$ that they generate is homogeneous....

**1**

vote

**2**answers

178 views

### Example of a projective variety over a field of characteristic zero which is uniruled but not ruled

A variety $ Z $ over a field $ k $ of characteristic zero is ruled if there is a variety $ M $ and a dominant, birational map $ \phi: M \times \mathbb{P}^{1}_{k} \dashrightarrow Z $. A variety $ Z $ ...

**12**

votes

**3**answers

908 views

### Motivation for the Jacobian Variety

I've been to many talks in Number Theory and for some reason I've yet to fully grasp, we all seem to like Jacobian Varieties a lot. I know that they are Abelian varieties, which give information about ...

**0**

votes

**0**answers

159 views

### A question on closed subschemes

Let $R \supset \mathbf{F}_q$ be a local ring with maximal ideal $\mathcal{m}$. We know that $\text{Spec}(R/\mathcal{m})$ is a closed subscheme of $\text{Spec}(R)$. Now, let $X$ be a smooth projective ...

**10**

votes

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359 views

### Is $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p$ coherent?

The question is as in the title:
Is the ring $\mathbb{Z}_p \otimes_{\mathbb{Z}} \mathbb{Z}_p = \mathbb{Z}_p \otimes_{\mathbb{Z}_{(p)}} \mathbb{Z}_p$ coherent?
As shown in the related question, the ...

**5**

votes

**1**answer

292 views

### Does the étale topos determine the Hodge numbers?

Does the small étale topos of a smooth proper variety over a perfect field of positive characteristic determine its Hodge numbers? We consider it as a Grothendieck topos over the étale topos of the ...

**5**

votes

**2**answers

429 views

### density of singular K3 surfaces

By singular K3 I mean a smooth complex K3 with Néron-Severi rank equal to 20.
Are singular K3 surfaces dense in the moduli space of polarized K3 surfaces?

**14**

votes

**1**answer

536 views

### Recover the characteristic of $k$ from the category of $k$-varieties

Can you recover the characteristic of a perfect field from the category of smooth projective geometrically connected varieties over it?

**2**

votes

**1**answer

160 views

### Polynomial constraints on the values of continuous functions $\mathbb{R}\to\mathbb{R}$

Denote by $S$ the set of closed points in $X=\mathrm{Spec}\:\mathbb{R}[x_\alpha]$ ($\alpha \in \mathbb{Q}$) that have $\mathbb{R}$ as their residue field. There is an injective map from the set of ...

**2**

votes

**1**answer

124 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 ...

**0**

votes

**0**answers

98 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, I'm studying how for an additive category $...

**13**

votes

**0**answers

297 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 $\mathbb{C}$?

**1**

vote

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87 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 ...

**1**

vote

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74 views

### Slope stability using the Riemann-Roch theorem [closed]

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 $ ...

**2**

votes

**0**answers

70 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 $1$, ...

**18**

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355 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 $$X\mapsto |X^{an}|$$ ...

**4**

votes

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112 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 $z_1$, ....

**16**

votes

**2**answers

545 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 ...

**2**

votes

**0**answers

172 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 ...

**6**

votes

**1**answer

202 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 \...

**1**

vote

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118 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)...

**6**

votes

**1**answer

224 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. ...

**3**

votes

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140 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 ...

**5**

votes

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65 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 ...

**2**

votes

**0**answers

107 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 ...

**4**

votes

**1**answer

243 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 ...

**2**

votes

**1**answer

145 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 ...

**1**

vote

**1**answer

155 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 ...

**2**

votes

**0**answers

88 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 ...

**4**

votes

**1**answer

151 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 ...

**1**

vote

**0**answers

64 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 ...

**2**

votes

**1**answer

265 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**

votes

**1**answer

167 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 $...

**7**

votes

**0**answers

107 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 ...

**1**

vote

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91 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 ...

**4**

votes

**0**answers

93 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 ...

**0**

votes

**0**answers

77 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 ...

**5**

votes

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136 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 ...