# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

14,774 questions

**2**

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103 views

### Divisibility of a divisor

Let $X$ be a smooth complex projective curve and $f \colon X \to Y$ an étale Galois cover, whose Galois group $G$ is finite and of order $r$. For any $g \in G$, define $$\Delta_g = \{(x, \, g \cdot x) ...

**3**

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**0**answers

95 views

### Functoriality of base change morphisms

Consider a commutative diagram of morphisms of schemes:
$$\begin{array}{ccccccccc}
X_2 & \xrightarrow{j_2} & Y_2 \\
f'\downarrow & & \downarrow f \\
X_1 & \xrightarrow{j_1} &...

**2**

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84 views

### Geometric irreducibility of fiber product of geometric irreducible schemes

Given three geometrically irreducible normal $k$-curves, $X$, $Y$, $Z$, and two morphisms $f \colon\ X \to Z$, $g \colon\ Y \to Z$. Assume that $X \times_Z Y$ is irreducible. Does $X \times_Z Y$ is ...

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112 views

### Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...

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74 views

### The moduli scheme of “$\nu$-canonically embeded curves”

This is related to the proposition 5.1. of Mumford's GIT.
It states that:
There is a unique subscheme $H$ in the Hilbert scheme $Hilb_{\mathbb{P}^n}^{P(x)}$ such that, for any morphism $f : S \to ...

**2**

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**0**answers

132 views

### De Rham cohomology and extension of scalars

Let $K$ be a field of characteristic zero and let $X$ be a smooth variety over $K$. Given a field extension $L/K$, I denote by $X_L$ the variety $X \times_{Spec(K)} Spec(L)$.
What is the easiest way ...

**7**

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**1**answer

212 views

### Embeddings of flag manifolds

Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...

**-1**

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**1**answer

209 views

### Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$

My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:
Given a projective variety $V$ defined over $L$ algebraically closed, of ...

**2**

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**0**answers

86 views

### Extending section of étale morphism of adic spaces

This question is related to Lifting points via étale morphism of adic spaces.
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...

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**1**answer

155 views

### Equivalence of definitions of Cohen-Macaulay type

I know that the Cohen-Macaulay type has this two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^...

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176 views

### The moduli scheme of smooth curves of given genus is irreducible

I've heard that Deligne-Mumford's "the irreducibility..." showed this first.
But I think that Mumford's "Geometric invariant theory" has its proof.
The proof is as follows:
Let $H$ be the scheme that ...

**3**

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**0**answers

86 views

### Automorphy Factor from Vector Bundles on Compact Dual

So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...

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76 views

### Reducible section of algebraic varieties

Let $X$ be a smooth algebraic variety and $Y$ a reducible linear section of $X$, then is it true (or under which assumptions it is true) that $Y$ is contained in a reducible hyperplane section of $X$ (...

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273 views

### When is a map of topological spaces homotopy equivalent to an algebraic map?

My question is simple, but I don't expect there are any simple answers.
Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....

**-2**

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148 views

### Injective map on spectra [on hold]

I know that if we have a surjection $f:B\rightarrow A$, this induces an injection on the spectra $f^* \colon \operatorname{Spec} A\hookrightarrow \operatorname{Spec} B.$
What about the opposite? ...

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143 views

### Atiyah class and coboundary map

Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $
be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j=...

**1**

vote

**0**answers

92 views

### Core of the Jordan quiver variety

It is known that, given the Jordan quiver, dimension vectors
$\textbf{v}=n,\textbf{w}=1$ and a stability condition $\theta<0,$ the corresponding quiver variety $\mathcal{M}_{\theta}(n,1)\cong \...

**1**

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140 views

### Level structures in deformation spaces of $p$-divisible groups

I am reading (parts of) the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein, and I am stuck at understanding the definition of level structures in Rapoport-Zink spaces (cf. Definition ...

**0**

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89 views

### When is geometric irreducibility an intrinsic property?

Let $k$ be a field. Let's say that $k$ has property $(∗)$ iff for a scheme $X$ and morphisms $f_1,f_2:X\rightarrow \mathrm{Spec}\,k$, geometric irreducibility of $f_2$ is implied by geometric ...

**1**

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**0**answers

95 views

### Density of $k$-rational points for non-algebraically closed field

Let $k$ be a perfect field.
We say that $k$ has property $(*)$ if for every geometrically irreducible morphism locally of finite type $X\rightarrow \mathrm{Spec}\, k$ admitting a section, the set of ...

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69 views

### Universal Property of the Zariski-Riemann Space

Let $k = \mathbb{Q}$ or $\mathbb{C}$. Let $K$ be a finitely generated field extension of $k$. A model of $K$ is a variety $V \subset \mathbb{CP}^n$ defined over $k$, such that the rational function ...

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99 views

### The Zariski Riemann Space, but with Local Rings

The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $\text{Spv}$ (the space of valuations) and Huber's work. In studying it, I would ...

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116 views

### Is there a $\sum e_if_i=n$ in higher dimensions?

If $X\to Y$ is a finite map of connected proper algebraic curves over a field, then for any point $y\in Y$, the sum $\sum e_xf_x=n$ of ramification times inertia degrees over points $x$ mapping to $y$ ...

**3**

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75 views

### Global functions on a flat proper family

Let $R$ be an integral domain. Let $f:X\rightarrow \mathrm{Spec}\,R$ be a flat proper morphism of schemes. Is it possible that $O_X(X)$ is not a flat $R$-module?

**4**

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**1**answer

137 views

### Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?

There is the following beautiful formula (see Qiaochu Yuan excellent blog):
$$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...

**5**

votes

**1**answer

124 views

### Orbits of action of the split group of type $F_4$

Let the split group of type $F_4$ act as the automorphism group of the split Albert algebra $A$. Consider the action of $F_4\times \mathbb{G}_m$ on $A$, given by letting $\mathbb{G}_m$ act by scalar ...

**3**

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50 views

### Metric with singularities on Riemann Surfaces and the associated Laplacians

I have asked this question on Math Stack Exchange
Metric with singularities and associated Laplacian
but I have not got any answers/comments, therefore I post this question on the MO.
Suppose $M$ ...

**0**

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**0**answers

55 views

### The algebraic variety behind a quotient of the ring of regular functions on the complex torus [closed]

I am trying to understand what algebraic variety stands behind the following quotient.
Let $I$ be the vanishing ideal of some linear subspace $\mathbb{C}^k \subset \mathbb{C}^n$ in the standard ...

**2**

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233 views

### Topological data of $K3\times T^{2}$

I need some help in order to clarify some topological data of a $K3\times T^{2}$ Calabi Yau manifold in which $K3$ part has been obtained as a resolution of a $T^{4}/ \mathbb{Z_{2}}$ orbifold .
EDIT:...

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157 views

+50

### On the algebraic cobordism cohomology of Eilenberg-MacLane spectrum

Let $MU$ be the complex cobordism spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum.
In this question it is shown that the group $MU^{*}(H\mathbb{Z})$ is non trivial.
From here, what can ...

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75 views

### Norm quadrics and their motives

Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...

**6**

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**1**answer

382 views

### Poincare duality on the level of complexes

The classical Poincare duality is formulated in terms of cohomology groups. I am wondering if we can also formulate it in terms of complexes.
In particular, suppose $\mathcal{C}^*$ is a complex of $...

**3**

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**2**answers

238 views

### Variety of conjugacy classes

Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...

**7**

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**0**answers

220 views

### Visualization and new geometry in higher stacks (soft question)

I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds ...

**7**

votes

**1**answer

130 views

### The relative dimension of blow-up and singularities

Let $X$ be an integral affine scheme of finite type over $\mathbb{C}$, $\mathrm{dim}\,X=d$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get ...

**2**

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**1**answer

188 views

### Étale fibration for $K[[X_1,…,X_n]]$

Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...

**3**

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**0**answers

94 views

### Computing affine Springer fibers

I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $G=SL_2$ over $\mathbb{C}$ and let $\mathcal{K}=\mathbb{C}((z))$ and $\mathcal{O}=\...

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205 views

### A Riemann Surface can be characterized by its Lie Algebra? [closed]

I was studying the geometric properties of Lie algebras.
Let $X$, $Y$ be two Riemann surfaces. Suppose that this surfaces meet a finite set of ordinary differential equations. Let $L_X$ and $L_Y$ ...

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**1**answer

141 views

### Semi-orthogonal decompositions for Calabi-Yau varieties

In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry there is an exercise (Exercise 1.7 and 1.8) to prove the statement that the derived category $D^b(X)$ of a Calabi-Yau variety $X$ has ...

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**0**answers

180 views

### Irreducible Smooth Proper one-dimensional Schemes isomorphic to $\mathbb{P}^1$

I have a curious question about an argument/hint given in following thread:
https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes
The OP asked if ...

**2**

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**0**answers

81 views

### Factoring Finite Morphisms through Open Immersions

This is a somewhat silly question, but I was wondering the following. Assume X, Y, and Z are schemes and consider the composition
$X \xrightarrow{f} Y \xrightarrow{g} Z $ and assume $g\circ f $ is a ...

**2**

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**0**answers

150 views

### Property of Complete Variety

I have a question about a step in the proof from Lang's "Abelian Varieties" (page 20):
By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible.
...

**1**

vote

**1**answer

103 views

### Projective subvarieties of blow-ups of affine varieties

Let $X$ be an integral affine scheme of finite type over a field $k$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a $k$-scheme $X'$. Is ...

**8**

votes

**1**answer

325 views

### Why is, for a group scheme of finite type, “smooth” (resp. irreducible) equivalent to “geometrically reduced” (resp. geometrically irreducible)?

I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479). Since I still don't have the permission to add images I quote ...

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**0**answers

97 views

### Is it possible for the Witt group of a scheme to have non-trivial odd torsion?

Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt ...

**7**

votes

**3**answers

415 views

### Borel's presentation for the cohomology of a Flag Variety

If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then
1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$
and
2) $K[T^\vee]^...

**4**

votes

**1**answer

125 views

### Automorphisms of singular hypersurfaces

Let $X\subset\mathbb{P}^{n+1}$ be an irreducible and reduced hypersurface of degree $d$.
A theorem by Matsumura and Monski asserts that if $n\geq 2$, $d\geq 3$, $(n,d)\neq (2,4)$ and $X$ is smooth ...

**3**

votes

**0**answers

152 views

### Shimura varieties of Hodge type

I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type".
I understand that ...

**9**

votes

**2**answers

318 views

### Nakano vanishing in positive characteristic

Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:
$(\ast) \quad$ $\mathrm H^...

**4**

votes

**1**answer

373 views

### Do higher etale homotopy groups of spectrum of a field always vanish?

Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $\mathrm{Spec}\,k$ vanish?
If the absolute Galois group is finite, we have a universal cover $\mathrm{Spec}\,k^...