# Questions tagged [ag.algebraic-geometry]

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

21,611
questions

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### A question related to the strong Oda conjecture

A fan is a collection of strongly convex rational polyhedral cones in $\mathbb Z^n$, which we often think of as contained in $\mathbb Q^n$ or $\mathbb R^n$ for purposes of visualizing it. The defining ...

3
votes

1
answer

261
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### Is the subscheme parametrizing the k-th degeneracy loci Cohen-Macaulay?

Let $X$ be a Cohen-Macaulay projective variety (say over an algebraically closed field of characteristic 0), and $E,F$ two vector bundles such that $E^*\otimes F$ is globally generated. Consider the ...

0
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1
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127
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### Double disk bundle

Double disk bundle: A smooth, closed manifold
$M \cong DB^{-} \cup_L DB^{+}$
where
· $B^{±}, L$ smooth, closed manifolds
· $D^{l± +1} → DB^{±} → B^{±}$ smooth disk bundles such that
$S^{l±} → L \cong ...

2
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0
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62
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### Existence of meromorphic differential form on curve with given multiplicity of zeroes and poles

Let $m \in \mathbb{Z}^n$ be a partition of $2g-2$. Polishuk showed in his paper "Moduli spaces of curves with effective r-spin structures" (arXiv link) that if all entries of $m$ are ...

2
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0
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137
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### Vanishing differential of Brown-Gersten-Quillen spectral sequence

Let $k$ be an algebraically closed field of characteristic zero and $X$ be an affine, simplicial toric 3-fold over the field $k$. I am trying to use the Brown-Gersten-Quillen spectral sequence to ...

2
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0
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75
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### Simply connectedness of leaves of a foliation on an complex manifold

Now I'm searching about leaves of foliation in the following special setting.
Let $U,V$ be two holomorphic vector field on $\mathbb{C}^2$ s.t the Lie bracket $[U,V]=UV-VU=0$ and $U$ and $V$ spaned ...

2
votes

1
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152
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### Identifying the circle bundle of the canonical line bundle $\mathcal{O}(-n-1)$ over a projective space $\mathbb{CP}^n$

It's not hard to see the following fact: the circle bundle of the tautological line bundle $\mathcal{O}(-1)\rightarrow \mathbb{CP}^n$ is $S^{2n+1}$, the unit sphere inside $C^{n+1}.$ I want to see ...

1
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0
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125
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### Blowing up $\mathbb{CP}^2$ nine times and exactness of symplectic form

Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in ...

2
votes

1
answer

197
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### Surjective étale map from simply connected curve over $\mathbb{C}$

Let $X$ be a simply connected algebraic curve over $\mathbb{C}$ and $f:X\rightarrow \mathbb{A}^1_{\mathbb{C}}$ is a surjective étale map.
Then is it true $f$ is finite?
All the domains of non finite ...

6
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0
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+50

### Is there a correction to the failure of geometric morphisms to preserve internal homs?

Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...

4
votes

1
answer

189
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### Fano scheme of cubic threefold in characteristic 3

Altman and Kleiman in Foundations of the theory of Fano schemes prove that if $F = F_1(X) \subset \mathbb G(1, 4) = G$ is a Fano scheme of lines on cubic threefold in $\mathbb P^4$ with at most ...

3
votes

1
answer

145
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### Pushforward of exceptional vector bundle is spherical for local P^2

I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical ...

1
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0
answers

114
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### What can we say when a module of differential is free?

Let $\mathbb{C}$ complex number.
$R=\mathbb{C}[x,y]/(f)(f\in \mathbb{C}[x,y])$
If the module of differential $\Omega_{R/\mathbb{C}}$ is free $R$ module of rank one,
what can we say about $R$.
How far ...

5
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0
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185
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### Applications of Langlands for GLn explicit reciprocity laws other than elliptic curves

Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$
for $n\ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves?
I've never found such ...

1
vote

0
answers

113
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### About the proof that deformation of canonical singularity is still canonical singularity

I was reading Professor Kawamata's paper Deformation of Canonical Singularity (which also appears in the book Algebraic Varieties: Minimal Models and
Finite Generation Corollary 2.12.11) the main ...

4
votes

0
answers

283
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### Merits of derived geometry

What are the merits of derived geometry? More precisely, which specific mathematical problems that can be formulated without this machinery have been solved using it? If those problems exist, could ...

16
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0
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300
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### Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )

Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...

4
votes

1
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171
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### Irreducible components of rigid varieties

I'm reading IRREDUCIBLE COMPONENTS OF RIGID SPACES (by Conrad). In this paper he defines the irreducible component of a rigid variety $X$ to be reduced image of a connected component of $\tilde X$ (...

3
votes

1
answer

133
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### On a multiplication map of divisorial sheaf

It maybe a very foolish question, but I can't prove or disprove this.
Suppose $X$ is a normal variety, and $L$ a $\mathbb{Q}$-line bundle on $X$. I.e., there is a $\mathbb{Q}$-Cartier Weil divisor $D$ ...

2
votes

1
answer

130
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### Understanding the integral $\int_0^1\det(v(t),v'(t))dt$ where $v(t)$ is path in the plane

Let $v(t) : [0,1]\rightarrow\mathbb{C}^2$ be a smooth path, and let $v' := dv/dt$. I'd like to understand what the integral:
$$I(v) := \int_0^1 \det(v(t),v'(t))dt$$
tells us about $v$, where $\det(v(t)...

2
votes

0
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143
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### On the definition of the relative canonical divisor

Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...

3
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0
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88
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### Vanishing cycles and component groups

Let $A$ be an abelian variety over a local field $K$ and assume it has toric reduction. Then two classical invariants associated to this are the component group $\Phi(A)=\mathcal{A}_s/\mathcal{A}_s^0$ ...

6
votes

1
answer

234
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### Criteria for when Gauss-Manin sheaves are vector bundles

Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...

3
votes

0
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319
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### The local global principle for differential equations

Are there any good reference to tackle the problem below?
Or, are there any know result?
Problem
Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector ...

4
votes

1
answer

227
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### Comparing Kummer maps to étale homotopy at finite level

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Hom{Hom}\newcommand{\et}{\mathrm{et}}\newcommand{\top}{\mathrm{top}}$In Voevodsky's paper "Étale topologies of schemes over fields of finite ...

1
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0
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101
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### Extension of MMP from the central fiber to some neighborhood

I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 )
There is a theorem about the extension of MMP step when the central fiber has ...

9
votes

1
answer

313
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### Maybe a folklore natural map between reflexive pullbacks

In the introduction of [HK04], it is proposed that for a morphism between varieties $f:X'\to X$, and a coherent sheaf $\mathcal{F}$ on $X$, there is a natural map $\alpha:f^*(\mathcal{F}^{\vee\vee})\...

3
votes

0
answers

161
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### A relative Abel-Jacobi map on cycle classes

I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations.
Background:
Suppose $X$ is a smooth projective ...

0
votes

0
answers

160
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### Why $k((x,t))$ can not be a local field?

If $k$ is a finite field, then $k((x))$ is a local field, and we can define a discrete valuation on $k((x))$ with respect to which it is complete. It is sometimes called a 1-dimensional local field.
I ...

5
votes

1
answer

249
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### Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...

0
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0
answers

122
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### Projective subvarieties are closed?

I want to show that projective subvarieties of a quasi-projective variety are closed. One possible solution should be the following:
Let $W \subseteq \mathbb{P}^n$ be a quasi-projective variety and $V ...

1
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0
answers

173
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### Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$.
Algebraically its $Spec$ is quite different from $k$. For example:
it has plenty non-trivial "line-...

3
votes

0
answers

155
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### What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?

Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...

3
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0
answers

92
views

### Existence of extensions of a flat projective morphism

Suppose that $S$ is a noetherian integral scheme and $U\subset S$ is an open subscheme. Let $f:X\to U$ be a flat projective morphism. I would like to know whether (or when) $f$ can be extended to a ...

2
votes

1
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175
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### Noetherian local ring with non-lci formal fibers

I am looking for an example of a noetherian local ring $A$ such that its formal fibers are not complete intersection rings in the sense of https://stacks.math.columbia.edu/tag/09PY (i.e., there is a ...

4
votes

1
answer

202
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### Equivariant projective embeddings with optimal dimension

Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find ...

4
votes

1
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162
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### Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?

I am currently reading Katz and Mazur’s Arithmetic moduli of elliptic curves and I am puzzled by a statement in the discussion of the $[\Gamma_0(N)]$ moduli problem in Chapter 3.
The authors define a $...

6
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0
answers

147
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### $K_0$ of arithmetic surfaces

In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated ...

20
votes

3
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669
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### Examples when quantum $q$ equals to arithmetic $q$

First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.
In the world of quantum mathematics, the letter $q$ is a standard ...

3
votes

1
answer

153
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### When is a del Pezzo surface a conic bundle?

I am considering over a field $k$ which is not algebraically closed, characteristic 0, and perhaps contains all the complex roots of unity that may appear. Feel free to realize it as some function ...

2
votes

0
answers

126
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### Classifying stack for finite flat group scheme

Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...

39
votes

1
answer

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### Clausen–Scholze's Theorem 9.1 of Analytic.pdf, in view of light condensed sets, AKA is the Liquid Tensor Experiment easier now?

In the recent lecture series run jointly from IHÉS and Bonn, Clausen and Scholze have reworked—again—their foundations of geometry to focus attention on not arbitrary condensed sets and solid modules ...

3
votes

1
answer

292
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### Does the absolute Frobenius induce the identity on étale topoi?

Let $X$ be a scheme defined over $\mathbb{F}_p$ and denote by $X_{et}$ its étale topos . Associated to $X,$ we can consider the absolute Frobenius map $F_X: X \rightarrow X$ which gives an associated ...

3
votes

1
answer

203
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### Etale cohomology of relative elliptic curve

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.
Let $R^1f_*\mathbb{Q}...

4
votes

2
answers

380
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### “Geometric” vs Homotopical completion

There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them.
The first one is the “homotopical” (or maybe it should be called ...

3
votes

1
answer

260
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### Is there a variety which is not locally set theoretic complete intersection?

A variety $X$ is a locally set theoretic complete intersection in a nonsingular variety $Y \supseteq X$ if each point in $X$ has a neighborhood $U$ in $Y$ such that $X \cap U$ is set theoretically the ...

6
votes

1
answer

395
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### Reference request: good reduction equivalent to crystalline étale cohomology

Suppose $X$ is an abelian variety over a $p$-adic field $K$, and it's well known that $X$ has good reduction is equivalent to the étale cohomology of $X$ is crystalline, and $X$ has semistable ...

1
vote

0
answers

93
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### One question about Manetti surface

I am reading Ascher-Devleming-Liu's paper "Wall crossing for K-moduli spaces of plane curves" theorem 5.2 ADL19 and l have some confusions about the proof.
Theorem 5.2 states that fixed a ...

0
votes

0
answers

78
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### Projectivity of equivariant K-theory of toric variety

I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups.
In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...

2
votes

0
answers

141
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### About pushforward of a sheaf of divisor

Let $X$ be a normal variety over an algebraically closed field of arbitrary characteristic, $f:X'\to X$ a log resolution, $L$ a Cartier divisor on $X$, and suppose $L\sim_{\mathbb{Q},f}E$, where $E$ ...