# Questions tagged [ag.algebraic-geometry]

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

18,832
questions

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### Stability of vector bundles and corresponding coherent sheaf

Let $j:Y\hookrightarrow X$ be an embedding of projective complex manifolds. Let $E\rightarrow Y$ be a vector bundle and $S=j_*E$ the corresponding coherent sheaf on $X$ (see Push forward of a Vector ...

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152 views

### Ind-etale vs weakly etale

In this article Bhatt and Scholze consider ind-etale and weakly etale maps of affine schemes. We have two (easy) statements, proven in Prop.2.3.3(1) and (5):
-- any ind-etale map is weakly etale,
-- ...

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227 views

### Are manifolds "naturally" ringed or locally ringed spaces?

My question is about whether it's more natural to see manifolds as ringed spaces or as locally ringed spaces. I think I have arguments for both points of view.
On the one hand, it's reasonable to ...

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229 views

### Birational morphism that is not successive blow-down along smooth centers?

Is there an example of a birational morphism of smooth complex projective varieties $f\colon X\to Y$, that cannot be factored into a chain $X\to X_1\to\cdots\to X_n\to Y$ of blow-down along smooth ...

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391 views

### Why is the $\operatorname{GL}_n$ character variety "cohomologically" the product of the $\operatorname{PGL}_n$ character variety and a torus?

$\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\GL{\operatorname{GL}}$This question is about an assertion in Mixed Hodge polynomials of character varieties, by Hausel and Rodriguez-Villegas. Fix positive ...

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95 views

### Enumerative geometry and restricted plane partitions

Donaldson-Thomas theory is an enumerative theory for virtual counts of ideal sheaves (with trivial determinant) of the structural sheaf $\mathcal{O}_{X}$ of some smooth projective manifold $X$.
There ...

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135 views

### The Lefschetz hyperplane theorem and when there is an even number of vanishing cycles

Let's suppose we have a smooth, complex algebraic surface $S \subset \mathbb{P}^N$ where $N$ is some large positive integer. Then, a generic Lefschetz pencil is a family of hyperplanes $H_t$ and we ...

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267 views

### Is there a relation on Hodge numbers, weaker than $h^{2,0}=0$, that implies a compact Kähler manifold is projective?

The Kodaira embedding theorem yields as a corollary that a compact Kähler manifold $X$ with $h^{2,0} =0$ is projective.
Is there a weaker relation on Hodge numbers that implies that a compact Kähler ...

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81 views

### Projectors and idempotents

An $n\times n$ matrix $P$ (over a commutative ring with identity $R$) is called a projector if $P^2=P$.
Let $X$ denote the $\mathbb{Z}$-affine subscheme of $\mathbb{A}^{n^2}$ that is defined by the ...

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163 views

### Coordinate ring of a flag variety

Edited:
[If G here is a simply connected semismple complex algebraic group.
A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$.
The ...

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73 views

### Algebraic hypersurfaces and Coxeter groups

What is the minimum degree of an algebraic hypersurface (not necessarily smooth) having each Coxeter group as its symmetry group?

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391 views

### Quasi-compact surjective morphism of smooth k-schemes is flat

I have precedently posted the same question on Math.Stackexchange (https://math.stackexchange.com/questions/4277856/quasi-compact-surjective-morphism-of-smooth-k-schemes-is-flat), but to no avail; I ...

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162 views

### Grothendieck group of admissible $p$-adic representations

Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\...

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1k views

### Reference request: Elementary proof of Lang's theorem

A few months ago, I read a nice elementary proof of Lang's theorem:
Theorem: Let $G$ be a connected linear algebraic group over $\overline{\mathbb{F}}_p$ and let $F : G \to G$ be a Frobenius map. Then ...

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252 views

### Number of rational points over finite fields mod $q$ is birational invariant

I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...

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305 views

### Noetherianity assumptions in Hartshorne's book

It seems that noetherian assumptions are not necessary in many results by Hartshorne, in his book "Algebraic Geometry". How much is this true? Could you please give examples?

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73 views

### If algebraic group $G$ acts faithfully on a $G$-qp variety $X$, then $G$ has a Faithful projective representation

In Michel Brion's survey on Linearization of algebraic group actions
is stated in Examples 3.2.2.(iv) following claim p 17
without proof:
We fix an algebraic group $G$ over field $k$ (of arbitrary ...

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71 views

### Flat $\mathbb{Z}$-lifts of Schubert cells and isomorphism for local cohomology groups

Let $G$ be a connected quasi-split reductive group over a field $k$ of characteristic zero with Weyl group $W$ and a Borel $B$.
Set $X:=G/B$, $C(w):=BwB/B \subset X$ for $w \in W$ the Schubert cell ...

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95 views

### $\mathbb{C}^*$-equivariant smooth completion of a quasiprojective variety

A famous theorem by Sumihiro states that, given a normal quasi-projective variety $X$ with a regular $G$-action (where $G$ is a connected linear algebraic group), there is a G-equivariant
projective ...

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241 views

### Number of roots over the rationals of a multivariate polynomial

Let $P(x_1,\dots,x_m)$ be a polynomial with $N$ roots over the rationals. If $N$ is finite, is there a known upper bound on $N$ in terms of $m$ and the degree $d$ of the polynomial? For $m=1$, an ...

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325 views

### $\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$

The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, this paper by Asok, Hoyois and Wendt. This of course implies storng $\mathbf{A}^1$-invariance results for the first ...

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125 views

### Ideal generated by a regular sequence

In Boocher and Grifo - Lower bounds on Betti numbers, in example 2.2 they say that if $R=k[x_1,\dotsc,x_n]$ is a polynomial ring and $M=R/(f_1,\dotsc,f_c)$ where $f_i$ form a regular sequence, then ...

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95 views

### Absolutely indecomposable objects and moduli space

In the setting of moduli spaces of vector bundles/quiver representations etc I've encountered a few times situations like the one that I'll try to explain thereafter .I was wondering if there's a &...

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134 views

### Do you know explicit examples of superelliptic curves $y^{\ell} = g(x)$ (for some prime $\ell > 3$) covering some elliptic curves?

For every elliptic curve $E$ Icart in $\S 2$ of the paper explicitly constructs a superelliptic curve $S\!: y^3 = f(x)$ and a cover $\varphi\!: S \to E$. Do you know explicit examples of superelliptic ...

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241 views

### Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?

We say a complex manifold $X$ has the resolution property if every coherent sheaf $\mathcal{M}$ on $X$ admits a surjection $\mathcal{E}\twoheadrightarrow \mathcal{M}$ by some finite rank locally free ...

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181 views

### Existence of genus 0 solution for linear ordinary differential equation

This question is about the linear differential equations with polynomial coefficients. I am interested in the necessary and sufficient conditions for the existence of genus 0 for linear differential ...

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93 views

### Does the Mukai's lemma hold for non-algebraic $K3$ surfaces?

In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry I found the following result due to Mukai (Page 232, Lemma 10.6)
Let $X$ and $Y$ be two $K3$ surfaces. Then the Mukai vector of any ...

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99 views

### $P^1$ bundle over complex tori

Let $M$ be a fiber bundle over $\mathbb C^2/ \Lambda$ whose fibers are $P^1$. $M$ is a complex manifold of dimensional 3. Is there a classfication about such $M$. And can we deduce that $M$ is a ...

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274 views

### What was supposed to appear in EGA after Chapter IV?

We find a nice table on the Wikipedia page mentioning for instance that abelian schemes were supposed to be discussed in Chapter XII.
Did anyone involved in EGA say anything more detailed, verbally or ...

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814 views

### A quotient space of complex projective space

Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...

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67 views

### Equivalence between smoothly regular and analytically regular

I think the following statement is true.
Let $M$ be a real analytic manifold. Let $S \subset M$ be an analytic or semianalytic subset. A point $p \in S$ is called smoothly regular resp. analytically ...

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115 views

### (Local) simple connectedness of irreducible algebraic varieties

Let $\mathbb k$ be an algebraically closed field of characteristic zero.
I have two questions:
(1) Is an irreducible algebraic variety $X/\mathbb k$ of dimension at least 2 locally simply connected?
(...

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195 views

### $\mathbb{Z}$-points in a given $\widehat{\mathbb{Z}}$-isomorphism class

Given a finite type $\mathbb{Z}$-scheme $X$ with $X(\widehat{\mathbb{Z}})\neq\emptyset$ can we find a finite type $\mathbb{Z}$-scheme $Y$ with $X\times \widehat{\mathbb{Z}}\cong Y\times\widehat{\...

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156 views

### Non-hereditarily locally transitive linear algebraic groups

I have encountered an algebraic group $G$ over $\mathbb C$ such that there is a Zariski open orbit for the adjoint action of $G$ on the nilpotent radical $\mathfrak n$ of its Lie algebra, but there is ...

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300 views

### Deformation invariance of Chern classes

Let $\pi:\mathcal X\to B$ be a deformation of a compact complex manifold $X=\pi^{-1}(0)$, then for any $t\in B$, the first Chern class $c_1(X)=c_1(X_t)$?
I know the Chern class of a manifold depends ...

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160 views

### do hyperfunction solutions always exist?

I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...

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127 views

### Is the quotient of a prestable curve by a finite group always a scheme?

Let $G$ be a finite group acting on a scheme $X$. It is shown in SGA1 (Expose 1, Prop 1.8) that a good categorical quotient exists (as a scheme) if and only if $X$ is a union of $G$-invariant open ...

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149 views

### Regularity of the Cartesian product of varieties

Let $U$, $V$ and $W$ be algebraic varieties of finite dimensions (in the case I am really interested, $U = \mathbb R$ and $V$ and $W$ are defined by a system of homogeneous polynomials in $\mathbb R ^{...

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632 views

### Could the Weil zeroes of curves be evenly distributed?

If $X$ is a smooth, geometrically connected, projective curve of genus $g$
over $\mathbb{F}_q$, then the zeta function of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of ...

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247 views

### Why geometric generic point (in abstract algebraic geometry) replace general points in the unit disk?

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
On $\mathbb{C}$ Lefshietz local results are as follows. Let $X$ be a ...

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341 views

### Are the quaternionic Grassmannians quaternionic Kaehler manifolds?

The complex Grassmannians $\mathrm{Gr}(n,r)$, of $r$-planes in $\mathbb{C}^n$ are Kaehler manifolds. What about the quaternionic Grassmannians of $r$-planes in $\mathbb{H}^n$ are they quaternionic ...

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250 views

### Perverse sheaves on the complex affine line

Show that a perverse sheaf on $\mathbb{A}^1(\mathbb{C})$ (the complex plane with the analytic topology) is a bounded complex $A$ of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology ...

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158 views

### Is the pull-back of canonical sheaf invertible (modulo torsion)?

Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K_X$ is not necessarily ...

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269 views

### Meaning of "the" general fiber in the paper "La conjecture de Weil : I"

In section 4.1, chapter 4 of Pierre Deligne's paper La conjecture de Weil : I (french version, translation to English) he states:
Let $X$ be a non singular analytic space and purely of dimension $n+1$....

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136 views

### Explicit construction of a presentation of a constructible sheaf of $\mathbb{Z}$-modules

This question was prompted by the two following:
Constructible étale sheaves on X are étale algebraic spaces over X
Naive question about constructing constructible sheaves
If I have a ...

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34 views

### Mapping a polynomial surface to another without "signature" change

This is probably easier to solve than my latest question and might be a useful lemma there.
Using the same notation as there, let $F(x,y,z)=0$ be a surface with degree $2$ in all $x,y,z$, and such ...

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109 views

### Projectivization of the normal bundle of $\mathbb P^4$ in the 10-spinor variety

Let $X=S_{10} \subset \mathbb P^{15}$ be the 10-dimensional spinor variety in its minimal embedding. Consider a $\mathbb P^4 \subset S_{10}$, hence we can define the normal bundle $N=N_{\mathbb P^4|S_{...

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131 views

### Canonical $\mathbb{Q}$-vector space for algebraic varieties over $\mathbb{Q}_p$

If we define $\mathbb{C}$ as the algebraic closure of $\mathbb{R}$ then we have a canonical map $\mathbb{R}\to \mathbb{C}$ and so for an algebraic variety $X$ over $\mathbb{R}$ we get a canonical $\...

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**2**answers

305 views

### Finite etale cover of projective line

If we have a map $f : \mathbb P^1_R \to \mathbb P^1_R$ over $\operatorname{Spec}(R)$, with $R$ a commutative ring, which we assume to be etale, then is it possible to characterize $f$? Must it be an ...

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**1**answer

212 views

### The numbers of isomorphism classes of abelian variety over finite fields

It is known that there are only finitely many isomorphism classes of abelian variety over a finite field. I am curious about the exact number of these isomorphism classes.
Explicitly, fix $g$, let $\...