# Questions tagged [ag.algebraic-geometry]

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

16,812
questions

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105 views

### Pushforward of the structure sheaf for smooth morphisms

Let $\pi:X\to S=\mathbb{P}^1_{\mathbb{Z}}$ be a smooth morphism. Is there a smooth separated morphism $\pi':X'\to S$ such that $\pi_*\mathcal{O}_X\approx \pi'_*\mathcal{O}_{X'}$ as $\mathcal{O}_S$-...

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46 views

### Uniform position for multiple components

(Modified from https://math.stackexchange.com/questions/3730261/uniform-position-theorem-for-reducible-varieties/3730457#3730457)
The uniform position theorem states (roughly) that a general ...

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185 views

### Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar.
Define an $\mathrm{SL}(2)$-action on $\...

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182 views

### A density criterion and a submersion map of a Hodge bundle

In Voisin's excellent book 《Hodge theory and complex algebraic geometry II》5.3.4 - a density criterion, there is a important theorem:
Let $X$ be a compact Kähler manifold, $\pi:\mathcal X \rightarrow ...

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**3**answers

870 views

### Are “large enough” finite etale covers arithmetic?

Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...

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109 views

### Scheme-theoretic image of the inverse image of a morphism of schemes

Let $f:X \to Y$ be a finite, surjective morphisms between noetherian, integral varieties (over $\mathbb{C}$). I am looking for conditions on $f$ under which I can say that for any closed subscheme $Z \...

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107 views

### pro-commutative group schemes

When $k$ is field, Demazure and Gabriel defined and worked with the category of commutative pro-algebraic groups over $k$. In their book, they proved that $Ext^n(\varprojlim G_i, H)= \varinjlim Ext^n(...

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219 views

### On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group

Background:
I've seen two versions of the homotopy exact sequence for etale fundamental groups. One from Stacks:
Stacks 0BTX: Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a ...

**3**

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**1**answer

115 views

### Are “strongly finite dimensional” homotopy invariant sheaves with transfers (locally) constant?

Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...

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144 views

### Riemann hypothesis for the motivic zeta function

To repair the failure of rationality in general (as shown by Larsen and Lunts for products of two curves of genus > 1) of M. Kapranov's zeta function defined for a variety over a field $k$ and ...

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156 views

### Why doesn't the Manin obstruction work for quadratic forms?

The Manin obstruction explains why the Hasse principle doesn't work for non-quadratic forms. Let's write the notation first;
$V(\mathbb{Q})$ is variety for rational numbers.
$V(A_\mathbb{Q})$ is ...

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208 views

### Galois action on torsion in homotopy groups not induced by homotopy equivalences

Let $V$ be a simply connected smooth projective complex variety defined over the rationals. Then for any integer $n\geq 2$ the group $\pi_n(V)$ is finitely generated abelian so profinite completion ...

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177 views

### Some computational results and goals of stable motivic homotopy theory of schemes

I am trying to learn ($\mathbb{P}^1$-)stable motivic ($\mathbb{A}^1$-)homotopy theory of schemes from the Cisinski-Deglise book, Triangulated Categories of Mixed Motives. In order to keep myself going ...

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179 views

### A homotopy equivalence from a variety to itself that is not homotopic to a homeomorphism

Let $V$ be a simply connected smooth projective complex variety. Can there be a homotopy equivalence $V\to V$ that is not homotopic to a homeomorphism?

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333 views

### Complex conjugation inducing a trivial map on the fundamental group

Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by ...

**14**

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**1**answer

270 views

### Degree of secant varieties of Veronese varieties

Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety.
Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...

**3**

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122 views

### Clemens-Griffiths component birational invariant

Let $X$ be a smooth variety over complex numbers $\mathbb{C}$, say a threefold for sake of better intuition. Is there any geometrical intuition behind the fact that
the Clemens-Griffiths component of ...

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**1**answer

101 views

### canonical divisor of a contraction

Let $X$ be a smooth quasi-projective variety and $Y$ be a positive dimensional subvariety. Let $Z$ be a variety obtained from $X$ by contracting $Y$.
My question is what is the relation between $K_X$ ...

**3**

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159 views

### Etale sites for stacks

Let $X$ be an algebraic stack, let $U\to X$ be a smooth cover by an algebraic space. In this setting, we have the big étale site of $X$ (if $X$ is a stack over a scheme $S$, this is the restriction of ...

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75 views

### Image of morphism of locally of finite type has a closed point?

Let $S$ be finite dimensional locally Noetherian regular scheme. Let $f \colon X \rightarrow S$ be locally of finite type.
Then $f(X) \subset S $ contains a closed point of $S$?
If it does, I'd like ...

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395 views

### Generalization of Weak Nullstellensatz?

I believe the following is standard, namely when $k = \bar{k}$ is algebraically closed there is a bijection between points and maximal ideals
\begin{eqnarray*}
k^n &\longrightarrow& \...

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266 views

### Stably trivial non-trivial vector bundles

I have two related questions. Can there be a stably trivial non-trivial holomorphic vector bundle over a closed complex manifold? Can there be a stably trivial non-trivial algebraic vector bundle over ...

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412 views

### Building all holomorphic vector bundles from the tangent bundle

Let $V$ be a smooth projective complex variety such that the canonical bundle is not trivial. We can construct some vector bundles over $V$ by starting with the tangent bundle and applying tensor ...

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51 views

### Commutative square of module of differential is cartesian?

Let $R$ be a regular local $\mathbb{Q}$-algebra and $f$ be a normal crossing divisor(i.e. $f = x_{1}x_{2}...x_{r}$ such that $R/(x_{i})$ is regular for each $i$). Then we have commutative diagram
$\...

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180 views

### Involution action on Brauer group of an abelian variety

Let $k$ be an algebraically closed field of characteristic $p>2$, let $A/k$ be an abelian variety. Let $\iota\colon A\to A, a\mapsto -a$ be the natural involution. Let $x\in\mathrm{Br}(A)[p]$ be a ...

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165 views

### Any finite flat commutative group scheme of $p$-power order is etale if $p$ is invertible on the base

This question is immediately related to Discriminant ideal in a member of Barsotti-Tate Group
dealing with Barsotti–Tate groups and here I
would like to clarify a proof presented by
Anonymous in the ...

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130 views

### Sheaf of smooth functions and restriction to a divisor

My question is targeted towards a very particular detail in my research that I am trying to understand. I will therefore break it down into some more general questions.
Let $X$ be a smooth variety, $i:...

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244 views

### Sheaf cohomology commutes with colimits of sheaves

Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof ...

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108 views

### Polynomial isometries of $\mathbb{A}^2_\mathbb{C}$

I have the following question, which I'm sure must be explored somewhere.
Consider a group of polynomial automorphisms of $\mathbb{A}^2_\mathbb{C}$ preserving a standard hermitian metric. Is there any ...

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74 views

### versal deformation ring of a p-divisible group with some tensors

I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with ...

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175 views

+150

### Interpolating between curves in different characteristics

Let $p\neq q$ be two primes. For a given integer $g>0$ choose a smooth proper geometrically connected curve of genus $g$ over $\mathbb{F}_p$ and similarly for $q$. Is there a proper flat morphism $...

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146 views

### hypersurface of degree d Hilbert polynomial

I am in trouble to solve part (2) Exercise 1.13 from "Moduli of Curves"
by Harris and Morrison on page 9:
Exercise (1.13)
2) Fix a subscheme $X \subset \mathbb{P}^r$. Show, by taking ...

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180 views

### Multiplication by $n$-map on group scheme

Let $G$ be group scheme over field $k$ and $n \in \mathbb{N}_{\ge 2}$ with $n \neq 0$ in $k$. Then $n$ induces a map $[n]: G \to G$ symbolically called "power by n" map. I think that the ...

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73 views

### Compactification of Picard variety over normal, projective varieties

Let $X$ be a normal, projective, integral variety (over $\mathbb{C}$) and $P$ be the Picard variety parametrizing invertible sheaves on $X$. Does there exist a compactification $\overline{P}$ of $P$ ...

**2**

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166 views

### Arnold-Liouville's theorem

Let $H:M\rightarrow B$ be a algebraically complete integrable system. Then Arnold-Liouville's theorem says that if the generic fibre is proper/compact then it must be an abelian variety.
But if the ...

**2**

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**1**answer

184 views

### $f^!=f^*[d]$ for quasismooth maps?

Given a smooth map of schemes $f:X\to Y$ of relative dimension $d$, then there is a natural isomorphism $f^!\simeq f^*[d](2d)$ (in any context where the six operations are defined; see Cesinski-...

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98 views

### Factorizing vector fields near manifolds of singularities

Let $V: \mathbb{R}^n \to \mathbb{R}^n$ be a smooth vector field containing a smooth $k$-dimensional manifold $M$ (with $1\leq k < n$) of singularities: $V(M)=0$.
Suppose furthermore that at every ...

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**1**answer

147 views

### Functoriality of Ext-functor

Let $X$ be a normal, integral variety and $U \subset X$ an open subset such that the complement of $U$ is of codimension at least $2$. Let $F$ be a coherent sheaf on $X$ such that $\mathcal{E}xt^1_U(F|...

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200 views

### Calculation in prismatic cohomology

In the standard references for prismatic cohomology, most theorems are proved in a local context (i.e. with completeness assumptions), and the devissage to the global case (i.e. smooth proper ...

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66 views

### A structure sheaf for real analytification of semialgebraic sets in the context of signed tropicalization

Let $X=Spec(A)$ be an affine scheme, where $A$ be a commutative algebra over a non-archimedean valued field $K$. Assume that $K$ is a real closed field with the unique ordering $<$, which should be ...

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142 views

### Profinite completion of the etale fundamental group and homotopy exact sequence

Let $X$ be a geometrically connected variety over a filed k; consider the exact sequence of etale fundamental groups
$$1 \rightarrow \pi_1 (X_{\bar{k}},\bar{x})\rightarrow \pi_1 (X,\bar{x})\rightarrow ...

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68 views

### Log point and monoid structure

My question is related to the definition of the standard Log Point.
It is defined in the following way: Let $k$ denote any field of $Char ~~0$ and $\mathbb N$ be the monoid of Integers. Then the map $\...

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74 views

### A complex analytic interpretation of multiplicity on the special fiber of a flat family

Let $X$ be a variety over $\mathbb C$ and $\pi: X\to \Delta$ be a flat morphism over the unit disk $\Delta=\{z:|z|<1\}$. Let $Z$ be a component of $X_0=\pi^{-1}(0)$. The multiplicity of $Z$ is ...

**0**

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**1**answer

76 views

### On relating $l(A), l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve where one of the divisors is effective

Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $k(X)$ denote its function field.
If $A, B$ are Weil divisors on $X$ such that $A$ is effective (i.e. $A\ge 0$) , then ...

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**1**answer

155 views

### A question about Dedekind schemes and proper morphisms

The following is Exercise 15.3 of Görtz-Wedhorn Algebraic Geometry I:
Let $S$ be a Dedekind scheme with function field $K$ and let $f: X\to S$ be a proper morphism of schemes. Then the canonical map $...

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**1**answer

329 views

### Results in “generalised smooth spaces” that did not hold in the case of smooth manifolds

Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page:
Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.
I understand that this category $\text{...

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51 views

### When minimal prime ideals are maximal with respect to not containing an element

Let $\{ P_i \}$ be the set of all minimal prime ideals of a commutative ring $R $. Is there any conditions on $R $ under which there exists an element $x\in R $ such that $P_i $ is an ideal of $R $ ...

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252 views

### Global functions on smooth varieties

Let $k$ be a field. Let $X\to \mathrm{Spec}\:k$ be a smooth morphism. Is there a smooth separated morphism $X'\to \mathrm{Spec}\:k$ such that $\Gamma(X, \mathcal{O}_X)\approx \Gamma(X', \mathcal{O}_{X'...

**14**

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**2**answers

396 views

### Direct image of the structure sheaf by an endomorphism of $\mathbb{P}^2$

Let us take 3 quadratic forms on $\mathbb{P}^2$ with no common zero; they define a map $\pi : \mathbb{P}^2\rightarrow \mathbb{P}^2$ of degree 4. It is not difficult to see that $\pi _*\mathscr{O}_{\...

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**2**answers

259 views

### Are Chow groups invariant under universal homeomorphisms?

Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...