# Questions tagged [ag.algebraic-geometry]

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

20,202
questions

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### complex K3 surfaces with automorphisms of given orders

Concerning complex K3 surfaces, there are various methods to show the non-existence of an automorphism of certain orders. The usually way is to investigate the action of the automorphism on the space $...

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32
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### Is it possible to extract a generalised criteria where an object has finite measure in (n+1) th dimension but has infinite measure in nth dimension? [closed]

Situation 1 : Koch's snowflake- Has infinite perimeter (Lets call it- 1D
measure) but finite area (Lets call it- 2D
measure)
Situation 2: Gabriel's horn- Has infinite area (Lets call it- 2D measure) ...

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### Basis of space of holomorphic differential of hyper elliptic curves $H^0(C,Ω_C/ \Bbb{Q})$

Let $C$ be an hyper elliptic curve and $J(C)$ be its Jacobean.
It is known that $H^0(C,Ω_C/ \Bbb{Q})$, space of holomorphic differential form on $C$ has basis(sometimes called 'Hermite basis') and can ...

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### Blow-ups of $ F $-regular varieties at points in general position and finite generation of the Cox ring

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

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### Is there a name for the number $ n $ of points in general position s.t. $ \operatorname{Cox}(\operatorname{Bl}_{p_{1},\dots,p_{n}}(Z)) $ is not f.g.?

Let $ Z $ be a projective, normal, $ \mathbb{Q} $-factorial variety (so the Cox ring of $ Z $ is well defined). Is there a name in the literature for the minimal natural number $ n $ such that the ...

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66
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### Derived subgroup of rational points vs. rational points of derived subgroups

Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion
$$
f: [G(k), G(k)] \rightarrow [G,G](k).
$$
If $k$ is not algebraically closed, $f$ is not necessarily ...

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### Converting an iterative function to calculate the trajectory of a particle, into a polar function

I have been working on trying to convert the below code (which simulates the trajectory of a particle [with air friction]) into a polar function that can be evaluated immediately (without having the ...

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1
answer

97
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### what is the linear system on a cubic surface giving the blow-down map to the plane

Consider $X$ a smooth cubic surface in $\mathbb{P}^3$, and let $l_1,...,l_6$ be six disjoint lines contained in $X$.
What is the linear system giving the blow-down map $X \to \mathbb{P}^2$, so that ...

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47
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### Space of valuations is spectral space and what does it mean to say that conditions are closed conditions

I am reading lecture 3 of Conrad notes (link : https://math.stanford.edu/~conrad/Perfseminar/ ), in which he proves space of valuations is a spectral space. Last theorem of lecture 3.
We have a map $j:...

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### Cartan–Remmert reduction of an algebraic variety

Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...

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### Do torsors $\phi: Z \to X$ (for varieties defined over a number field $K$) induce continuous maps $\phi: X(K_v) \to Z(K_v)$ in the v-adic topology?

Let $K$ be a number field, let $G$ be some linear algebraic $K$-group and let $\phi: Z \to X$ be a $G$-torsor over $X$. Let $v$ be a place of $K$.
Does the morphism $\phi$ induce a continuous map $\...

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1
answer

106
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### Invariant ring of the subvariety

Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $...

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### Examples of compact non-Kähler complex manifolds with Kodaira dimension zero

Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$.
Is there a known example where the canonical bundle is not holomorphically torsion?
For ...

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1
answer

107
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### Only finitely many rational curves in a complete linear system of a K3 surface

Consider a projective complex K3 surface $X$, then $\lvert D\rvert$ contains only finitely many rational curves for any divisor $D$ on $X$.
What is the original reference for this result?

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97
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### Moduli stack of doubly periodic complexes?

Let $\mathcal{A}$ be an abelian category.
In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...

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145
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### Faithfully flat descent in complex analytic geometry

A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "...

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292
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### Localic or topos-theoretic definition of $\operatorname{Spec}$

Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\operatorname{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes&...

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### Is the Cox ring of a $ \mathbb{Q} $-factorial, $ F $-regular, Mori dream space $ F $-regular?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

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+50

### Construction of a line bundle from a class $[\alpha] \in H^1(X, \mathcal{O}_X^{\times})$ as $\mathcal{O}_X^{\times}$-Torsor

Let $X$ be a complex compact manifold, and write $\mathcal{O}_X$ for the sheaf of holomorphic functions on $X$. Let $\mathcal{O}_X^{\times}$ be the subsheaf consisting of holomorphic functions. These ...

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vote

1
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243
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### Sheaf cohomology in number theory

I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...

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146
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### Approximate versions of Segre's Theorem

Consider projective $2$-space over a finite field of odd prime characteristic $p$. We say a set of points, $A$, in this space is an arc if any line meets it in at most two points. We say that an arc ...

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167
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### Bundles vs. line bundles

Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-...

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answer

212
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### About simple motives

I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...

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106
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### The Brauer group of the function field of a proper curve

Let $X$ be a smooth proper geometrically connected curve over a number field $k$, and let $k(X)$ denote its field of rational functions, i.e., its function field. Then the (cohomological) Brauer group ...

2
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answers

142
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### References to let me know about current directions of research in arithmetic geometry

I have knowledge of basic algebraic geometry and good deal of number theory. I have studied roth theorem and I am currently studying proof of Mordell-Weil theorem. These two topics come under ...

8
votes

1
answer

248
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### The Grothendieck topology of closed immersions on schemes

Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$...

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157
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### Monodromy group action on de Rham cohomology

Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...

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0
answers

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### Example of projective, $ F $-regular variety $ X $ and smooth sub-variety $ Y $ such that $ \operatorname{Bl}_{Y}(X) $ is not $ F $-regular?

A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...

5
votes

1
answer

331
views

### Six people standing on earth

Consider 6 people $p_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p_i$ again. Suppose no pair of these points $p_i$ are antipodal. At each point $p_i$ ...

2
votes

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194
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### Automorphism of moduli space of stable vector bundles over a curve

Let $C$ be a smooth genus two hyperelliptic curve and $\mathcal{M}_C$ be a moduli space of stable rank two vector bundles of fixed degree(or fixed determinant). Then is $\mathrm{Aut}(\mathcal{M}_C)\...

5
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174
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### Which properties of the pullback/restriction functor imply surjectivity of a ring homomorphism

Let $f:R\to S$ be a homomorhpism of (noncommutative) algebras over some field k (say the complex numbers) and let $F:Rep(S)\to Rep(R)$ be the corresponding functor between the categories of finite-...

5
votes

1
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150
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### How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point

I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
Setting
Let $\mathcal{M}_{1, 1, k}$ ...

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### Number of holomorphic line bundles with holomorphic sections for a fixed cohomology class

Let $X$ be a smooth compact Kähler (or more strongly projective) manifold and $\alpha$ an element of its Néron-Severi group. Let $\mathrm{Pic}^{\alpha}(X)$ denote the subset of the Picard group $\...

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185
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### Fourier-Mukai transform is the derived functor

In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me.
Let $X$ be an abelian variety over an ...

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60
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### Intersection of stabilizer group orbits and algebraic variety of decomposable forms

I have been trying to prove/come up with counter examples to the following situation, any help would be very much appreciated.
Let $\{E_I\}$ be a basis of $\mathbb R^6$, so that any vector $V\in\...

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0
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43
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### Is the Schofield semi invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...

0
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118
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### Cone of morphism induced by Serre duality

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...

4
votes

1
answer

139
views

### Normal bundle of a linear subspace

Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...

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votes

1
answer

161
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### Are there any non-elementary functions that are computable?

Does a function $\mathit{f}:\mathbb{R}→\mathbb{R}$ being non-elementary (not expressible as a combination of finitely many elementary operations), imply that it is not computable?
The particular case ...

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97
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### Pull back of ample divisor by a birational morphism

Let $X$ be a normal projective variety of Picard number one. Let $\pi: Y \to X$ be a resolution of singularity of $X$ and $A$ is the ample generator of $\text{Pic}(X)$. Then $\pi^*A$ is nef but not ...

3
votes

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175
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### Ample toric divisors

Let $X$ be a smooth complex projective variety (of dimension two or higher) and $D=\bigcup D_i$ be a simple normal crossings divisor on $X$ such that:
$D$ is ample (maybe we need very ample but I am ...

2
votes

1
answer

130
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### Find an analogue of Weyl chamber structure

Let $G$ be a split reductive group and let $T$ be a split maximal torus whose rank is $l$. Is it possible to find a base $\gamma_1,..., \gamma_l$ of the weight lattice $X(T)$ such that the cone $C$ in ...

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53
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### Semistability of the pull back of sheaf of logarithmic one-forms modulo the torsion under birational map

Let $X$ be a projective Fano variety ( need not be smooth) of Picard number one. Let $D$ be a reduced divisor on $X$ and $U$ be an open subset contained in smooth locus of $X$ such that $D \cap U$ is ...

0
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1
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166
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### Depth of almost complete intersection rings

Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\...

3
votes

1
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208
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### Siegel modular forms in Mathematica

Is there a convenient way to work with Siegel modular forms in Mathematica? I am interested in doing analytic computations using the $\chi_{10}(\Omega)$ Siegel modular form, where $\Omega$ is the $2\...

0
votes

1
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95
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### Compatibility conditions for quadratic equations

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$:
\begin{eqnarray}
0 &= A_1x^2 + B_1x + C_1 \\
&...

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0
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67
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### Intersection product of $\mathbb{Q}$-Cartier divisors with irreducible complete curves is well-defined

I am learning the notion of intersection product of a $\mathbb{Q}$-Cartier divisor with an irreducible complete curve on a normal variety. The definition I learned is that if $D$ is a $\mathbb{Q}$-...

2
votes

0
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100
views

### Results concerning surjectivity of Brauer groups

Are there known cases of a morphism of smooth geometrically connected curves $f: X \rightarrow Y$ over a number field $k$ (to be specific) that would give rise to a surjective restriction map $f^*:\...

4
votes

0
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148
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### Rank $2$ motivic local systems on a curve

This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam.
In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any ...

5
votes

1
answer

307
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### Geometric properties of the adjoint action of a reductive group

$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a reductive algebraic group over field $k = \overline{k}$ and consider the characteristic polynomial $\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$ induced ...