# Questions tagged [ag.algebraic-geometry]

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

17,182
questions

**3**

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61 views

### how much information $O_K$ points of a formal scheme over $\mathbb{Z}_p$ contain

assume that $Spf\,A\to Spf\,\mathbb{Z}_p[[t_1,...,t_n]]$ is a closed immersion of flat integral formal schemes over $\mathbb{Z_p}$. I see Kisin several time use that if $Spf\,A(O_K)\subset SPf\,\...

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69 views

### Unibranch points (definition for varieties over arbitrary field)

In David Mumford's book Algebraic Geometry I, Complex Projective Varieties
treating mainly complex varieties as objects of interest on page
43 he defines what is a topologically unibranch variety $X$ ...

**3**

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126 views

### Is there bijective correspondence among the following $3$ sets?

Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. Consider a power series $f(x) \in O_K[[x]]$ ...

**5**

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137 views

### Serre: Duality of regular differentials

I already asked this on math.stackexchange.com, but didn't get any responses. I hope it is appropriate here.
Let $X'$ be an irreducible singular algebraic curve over an algebraically closed field $k$, ...

**3**

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**1**answer

92 views

### Fibers of Hitchin fibration are equidimensional

Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\ge 3$, $M$ be a moduli space of stable vector bundles on $X$ of rank $n\ge 2$ and degree $d$, $\mathcal{M}$ be a moduli space of ...

**2**

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36 views

### What are possible applications of 'fast arithmetic' in the Jacobian (degree zero Picard group) of projective curves over fields?

It is well known that there are plenty possible applications of 'fast arithmetic' (that is, 1. having an algorithm at hand that actually computes in..., and 2. the running time of that algorithm is ...

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68 views

### Cotangent bundle of moduli space of stable bundles

I think this is a basic and dumb question.
Let $X$ be a smooth projective curve over $\mathbb{C}$, $M$ be a moduli space of stable bundles and $\mathcal{M}$ be a moduli space of semistable higgs ...

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76 views

### Variation of Morse Functions: a reference request

Suppose I have a manifold $X$ and a family of Morse functions $F_t:X \times \mathbb R \to \mathbb R$ on it where $t$ is the second parameter. So, if we fix $t$, we get a regular Morse function for ...

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54 views

### Polarization of Prym varieites

I'm trying to understand polarization and rational Hodge structure of spectral curves and Prym varieties.
Excuse me that this is similar to my previous question.
I want to prove the following,
Let $X$...

**8**

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146 views

### Confusion on basic example of homological projective duality

I've been trying to learn more about Kuznetsov's homological projective duality, and I'm confused about a remark in his notes for his ICM address. At the end of section 4.1, he says:
In particular, ...

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102 views

### computational complexity of rational parametrisation

Let $\cal C$ be a space curve defined as the common zeros of $n-1$ polynomials. Assuming that $\cal C$ admits a rational parametrisation, there is an algorithm generating a point on the curve through ...

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145 views

### Leech lattice and rational varieties

Question: Is there a smooth rational variety $X$ of complex dimension $4n$, $n \in \mathbb{N}$; such that the intersection form on $H^{4n}(X,\mathbb{Z})$ is the Leech lattice?
My motivation is mainly ...

**6**

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**4**answers

188 views

### Blow up of projective variety $P^1 \times P^1… \times P^1 (n $ times) and Blow up of $P^n$

It is known that Blow up of $P^1 \times P^1$ at a point is isomorphic to Blow up of $P^2$ at two points. Wondering if there is any general statement for Blow up of $P^1 \times P^1 \times.... \times ...

**6**

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**0**answers

128 views

### nonvanishing higher cohomology of a very ample divisor

I am looking for smooth projective varieties $X$, with $h^i(X, \mathcal{O}_X) = 0$ for $i > 0$, with a very ample line bundle $L$ with some nonvanishing higher cohomology.
What is clear:
(1) Curves ...

**4**

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**0**answers

124 views

### Can we attach (formal) abelian varieties to $p$-adic modular forms?

The Jacobian of the modular curve $X_1(N)$ over $\mathbb Q$ $J_1(N)$ can be decomposed up to isogeny, as a product of abelian subvarieties $A_f$ corresponding to Galois conjugacy classes of Hecke ...

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139 views

### Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?

By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?

**5**

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87 views

### Conjugacy classes of plane k-jet group

Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(...

**6**

votes

**3**answers

421 views

### Spectrum of a ring (studied by Krull?) of rational functions

Let $k$ be an algebraically closed field and $\mathbb A^2_k=\operatorname {Spec}k[x,y]$ the affine plane over $k$.
Consider the ring $R \subset k(x,y)$ of the rational functions on the plane defined ...

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100 views

### The compactified Jacobian is birational to a $\mathbb{P}^1$-fibration over the Jacobian of normalization

Let $Y$ be an integral curve whose only singularity is one simple node at a point $y$, and
$\pi:X\rightarrow Y$ be the normalization with $\pi^{-1}(y)=\{x,z\}$. $J(X)$ is the Jacobian of $X$, and $\...

**2**

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**1**answer

153 views

### Extension of a holomorphic vector bundle on a nodal curve

I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve.
Let $C$ be a nodal curve without closed componets and $E$ a ...

**9**

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**1**answer

386 views

### What is the automorphism group of the projective line minus $n$ points?

$\DeclareMathOperator{\AGL}{\operatorname{AGL}}\DeclareMathOperator{\PGL}{\operatorname{PGL}}$What is the automorphism group of $\mathbb P^1$ minus $n$ points (let's say over an algebraically closed ...

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100 views

### Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$

Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...

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**1**answer

94 views

### Varieties in the $\mathit{GL}(V)$-module $S_{\lambda}V$

It is known that, given a complex vector space $V$ of dimension $\mathit{dim}(V)=n+1$, all irreducible representations for the group $\mathit{GL}(V)$ are parametrized by Young tableaux $\lambda$.
For ...

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57 views

### Minimum number of generators of the product of ideals

Let $k$ be an algebraically closed field, let $I, J \subseteq k[x_1,\dots, x_n, y_1,\dots, y_m]$ be ideals, and let $i,j$ be the minimum number of generators of $I$ and $J$, respectively. I have two ...

**6**

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**1**answer

141 views

### Strict transform of a tangent curve under blow-up

$\DeclareMathOperator{\Bl}{\operatorname{Bl}}$It is known that if we have a projective variety $X$ and a projective smooth subvariety $Y$ then the exceptional divisor $E \subset \Bl_{Y}X$ of the blow-...

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86 views

### Fiber of the Hitchin map

Let $C$ be smooth projective curve, $\mathcal{M}$ be the moduli space of semistable Higgs bundles, $H:\mathcal{M}\rightarrow W= \bigoplus H^0(C,K_{C}^{\otimes i})$ be the Hitchin map, and $\pi :C_s\...

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60 views

### Generalizations of a theorem of Springer to other varieties

Let $F$ be a complete discretely valued field with ring of valuation $R$, uniformizer $\pi$, and residue characteristic $\neq 2$.
A theorem of Springer says that a quadratic form $q=q_1 \bot \pi q_2$, ...

**2**

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**1**answer

107 views

### References for relative ext-sheaves

I would like to know if there are references for relative ext-sheaves. The only one that I have is the paper of H. Lange (Universal families of extensions, Journal of Algebra 83, 101-112 (1983)), and ...

**4**

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109 views

### Quasi-unipotent monodromy for variation of Landau-Ginzburg cohomology

A pair, $(X,f)$, consisting of a smooth variety and a global function $f:X\rightarrow\mathbb{A}^{1}$ is called a Landau-Ginzburg model, LG-model for short. The LG-cohomology of the pair, dentoed $H(X,...

**6**

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**1**answer

313 views

### Are the Galois actions on automorphisms of twists isomorphic?

This might be a trivial question and I might be overlooking something:
Suppose $k$ is a field with algebraic closure $\overline k$ and absolute Galois group $\Gamma$. Let $X,Y$ be two distinct ...

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33 views

### Calculating the circumference of a circle formed from a linear distance [closed]

If you took a linear distance of 10 m and wrapped it around to form a circle would the circumference of the circle be 10 m and ,if not, why not or what would it be? Is there any equation to do this ...

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150 views

### Almost ring theory and derivations

I don't understand the definition of $\boldsymbol{\Omega}_A$ in the context of almost rings. In Gabber and Ramero https://arxiv.org/pdf/math/0409584.pdf it is covered in 9.6.12. How is $\boldsymbol{\...

**6**

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184 views

### Strict Henselization vs base-change to algebraic closure

Let $x$ be a smooth $k$-point on a variety $X$ over a field $k$ of characteristic $0$.
Is the strict Henselized local ring $\mathcal{O}_{X,x}^{\mathrm{sh}}$ the same as $\mathcal{O}_{X,x}^{\mathrm{h}} ...

**4**

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86 views

### Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$

Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...

**6**

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151 views

### Geometric foundation of the Grothendieck polynomials

Grothendieck polynomials were firstly defined in
Alain Lascoux and Marcel-Paul Sch¨utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux....

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97 views

### The “easier” case in Knudsen's stabilization of pointed curves

My question concerns Knudsen's (super-important) proof that $\overline{M}_{g,n+1}$ is the universal curve over $\overline{M}_{g,n}$ (F. Knudsen, The projectivity of the moduli space of stable curves ...

**6**

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77 views

### Toric description of tautological bundle associated to a weighted projective space

Let $a_1,\ldots,a_n\in\mathbb{N}$ be pairwise coprime positive integers, and consider the weighted projective space $$\mathbb{P}(a_1,\ldots,a_n).$$
It is well known that a weighted projective space is ...

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68 views

### generalizations of semi-continuity theorems

I would like to know some references (if any) about the generalization of the semicontinuity theorem and "Cohomology and base change" theorem (theorems 12.8 and 12.11 in the book "...

**3**

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112 views

### Cohomology of Siegel modular varieties

$\mathcal{A}_g(N)$ is the moduli space of principally polarized abelian varieties with a level $N$ structure.
Set $C_g=\displaystyle{\lim_{\rightarrow}} H^3(\mathcal{A}_g(N), \mathbb{F}_p)$ where the ...

**3**

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74 views

### Using principal polarisation to “cancel” Jacobian summands in isomorphism

I'm working through the sketch proof of irrationality of cubic threefolds in Huybrechts' The geometry of cubic hypersurfaces.
Let $J(X)$ denote the intermediate Jacobian of a cubic threefold $X \...

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162 views

### Moduli of finite-dimensional algebras

Let $n\geq 1$ be an integer. There is an obvious family of $n$-dimensional unital algebras parametrized by $\mathbb{C}^{n(n-1)^2}$ such that any $n$-dimensional unital algebra is isomorphic to at ...

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71 views

### Computing the Hilbert series of an irreducible component of a complete intersection

There's a nice formula for the Hilbert series of any complete intersection of hypersurfaces $X_1\cap\cdots\cap X_i\subseteq\mathbb{P}^n$ in terms of the degrees of $X_1,\ldots,X_i$. Is there a way to ...

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313 views

+50

### A question about pullbacks of $C^\infty_M$-modules

First, let me state a definition: Let $M$ be a smooth manifold and suppose $\mathcal{E}$ is a sheaf of $C^\infty_M$-modules. Given a point $x \in M$ let $I_x$ denote the vanishing ideal at $x$. We ...

**3**

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70 views

### Cohomology with supports of dualizing sheaf

Let $Z$ be a closed subvariety of $X$ and $T_{Z/X}$ the relative tangent complex. Possibly I want to assume that $Z \hookrightarrow X$ is a regular embedding, so that $T_{Z/X}$ is just a shift of the ...

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46 views

### Natural correspondence between the set of morphisms and the set of global sections

I'm trying to prove the following claim.
Let $X$ : smooth projective scheme over $\mathbb{C}$, $L$ : line bundle over $X$ and $A$ : $\mathbb{C}$-algebra.
$X_A=X\times_{\mathbb{C}} \operatorname{Spec}A$...

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**1**answer

118 views

### Divisors on the symmetric product of an elliptic curve

Assume that $C$ is an elliptic curve and $C_p$ is the $p$-fold symmetric product. Let $\beta:C_p\to C$ be defined by the addition on the elliptic curve. Let $u\in C$ be the zero in the additive group ...

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31 views

### The data of bundle structure of $\operatorname{End}_{0}E$ with Lie algebra

This may be a trivial question. I want to show the following claim about the structure of Lie algebra bundles,
Let $X$ be a smooth projective curve over $\mathbb{C}$, and $E$ be vector bundle over $X$...

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195 views

### Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?

In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...

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86 views

### The isomorphism classes of lifts of a $\operatorname{PGL}_r$-bundle to $\operatorname{GL}_r$

I want to show the following Lemma,
The set of algebraic isomorphism classes of lifts to $\operatorname{GL}_r$ of an algebraic principal $\operatorname{PGL}_r$ -bundle $P$ on a smooth projective ...

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127 views

### Perverse restriction

Let $f:U\to V$ be a separable dominant morphism of irreducible positive-dimensional varieties. Let $F$ be a perverse sheaf on $U$. Are there infinitely many closed points $p\in V$ such that $F|_{U_p}$ ...