Questions tagged [ag.algebraic-geometry]

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

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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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3answers
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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0answers
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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0answers
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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1answer
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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1answer
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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0answers
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 ...
3
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0answers
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 ...
5
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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 ...
3
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0answers
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?
5
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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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1answer
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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0answers
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 ...
1
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1answer
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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0answers
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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0answers
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 ...
9
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3answers
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& \...
5
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1answer
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 ...
5
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1answer
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 ...
2
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0answers
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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1answer
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 ...
2
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0answers
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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0answers
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:...
4
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2answers
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 ...
1
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1answer
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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0answers
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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0answers
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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0answers
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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0answers
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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0answers
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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1answer
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-...
2
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0answers
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 ...
1
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1answer
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|...
5
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0answers
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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0answers
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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0answers
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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0answers
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 $\...
2
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0answers
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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1answer
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 ...
0
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1answer
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 $...
8
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1answer
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{...
1
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0answers
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 $ ...
3
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0answers
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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2answers
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}_{\...
4
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2answers
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 ...

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