Questions tagged [ag.algebraic-geometry]

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

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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 ...
3
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0answers
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, -- ...
3
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0answers
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 ...
6
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1answer
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 ...
8
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1answer
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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0answers
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 ...
3
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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 ...
6
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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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2answers
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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2answers
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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1answer
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 ...
2
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1answer
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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0answers
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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1answer
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 ...
2
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0answers
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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0answers
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 &...
5
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1answer
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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1answer
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 ...
5
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1answer
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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0answers
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 ...
11
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3answers
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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0answers
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 ...
2
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0answers
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? (...
6
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0answers
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{\...
6
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1answer
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 ...
4
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1answer
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 ...
3
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1answer
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 ...
3
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0answers
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 ...
0
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1answer
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 ^{...
6
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2answers
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 ...
2
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1answer
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 ...
5
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1answer
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 ...
4
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1answer
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 ...
4
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1answer
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 ...
3
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0answers
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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0answers
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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0answers
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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0answers
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_{...
3
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0answers
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 $\...
3
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2answers
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 ...
2
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1answer
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 $\...