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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4
votes
1answer
152 views

Étale covers pulling back a very ample class to any integer multiple

Let $V$ be a smooth complex projective variety. Choose a very ample class $H\in H^2(V, \mathbb{Q})$. Can there exist finite étale morphisms $\phi_k:V\to V$ for each $k\geq 1$ such that $\phi^*_kH=kH$?
4
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0answers
139 views

Tannakian group of Galois representations coming from geometry

Let $K$ be a number field. Let $G_K$ be its absolute Galois group. Let $p$ be a rational prime. Let $\mathcal{R}_{K,p}^g$ be the category of finite-dimensional continuous $p$-adic representations of $...
2
votes
0answers
40 views

Linearly dependent points and the uniform position theorem

One proof of the uniform position theorem (as stated in p. 109 or p. 113 in Section III.1 of "Geometry of Algebraic Curves") uses a monodromy argument. While this gives us something even ...
0
votes
0answers
61 views

Locally freeness of twisted direct images $\pi_* \mathcal{F}(i)$

I have a question about a step in the proof of the Existence of Flattening Stratification I found in Nitsure's paper here: https://arxiv.org/abs/math/0504590 This question is closely related to Local ...
6
votes
0answers
121 views

Completion/Compactification of a Kähler metric on $\mathbb C^2$

Consider $\mathbb{C}^{2}$ equipped with the Kähler form $$ \omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right), $$ where $\mu$ is a positive real ...
5
votes
1answer
291 views

Is the action of the absolute Frobenius on de Rham cohomology induced by an algebraic map?

Let $X\to \mathrm{Spec}\:\mathbb{Z}_{(p)}$ be a smooth proper morphism with a geometrically connected generic fiber. Assume that the special fiber has an $\mathbb{F}_p$-point. Via the isomorphism $H^{*...
8
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0answers
239 views
+250

Reconstruct a variety from its crystalline topos

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point. Can we reconstruct $X$ from its small crystalline topos $((X/...
6
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0answers
160 views

Arithmetic schemes with the same zeta function

Suppose $X$ and $Y$ are $n$-dimensional regular separated schemes of finite type over $\mathbb{Z}$ such that number of $\mathbb{F}$-points of $X$ and $Y$ are equal for all finite fields $\mathbb{F}$. ...
4
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0answers
230 views

Étale cohomology and normalization?

I have an argument, which I wonder if someone could check: Let $X$ be an irreducible reduced scheme over a field $k$. Then we have a normal scheme $X^{norm}$ with a finite birational $f:X^{norm}\...
4
votes
1answer
173 views

Pushout of schemes and étale cohomology

Let $k$ be an algebraically closed field and $X,Y$ two $k$-schemes. We fix a $k$-point in $X$ and in $Y$ each, which we denote by abuse of notation by $P$. Since the pushout of schemes along closed ...
4
votes
2answers
276 views

Katz's proof of Cartier's (descent) theorem

I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
0
votes
0answers
86 views

Subsheaves of constant sheaves

Let $X$ be a connected topological space. I am looking for examples of a locally constant subsheaf (of $\mathbb{C}$-vector spaces) of a constant sheaf (of $\mathbb{C}$-vector spaces) on X, which is ...
2
votes
0answers
98 views

Anabelian Ehresmann and deformation invariance of $l$-adic Chern classes

Let $S$ be a connected scheme of finite type over $\overline{\mathbb{F}_p}$. Let $\pi:X\to S$ be a smooth proper morphism such that each fiber has a trivial étale fundamental group. Let $s, s'\in S$ ...
4
votes
1answer
139 views

Deformation equivalent varieties over an irreducible base

Fix an algebraically closed field $k$. Let $X$ and $Y$ be proper varieties over $k$. If there is a connected scheme $B$ of finite type over $k$ such that $X$ and $Y$ embed in a proper flat family over ...
1
vote
1answer
108 views

Is a ideal of direct limit of rings is itself a direct limit of ideal?

By Ńeron-Popescu desinaglarization theorem, If R is any regular semi-local ring containing $\mathbb{Q}$. Then R is a direct limit of regular semi-local rings $R_{i}$, where each $R_{i}$ is essentially ...
4
votes
1answer
173 views

Blow up the diagonal of a symmetric product space

Consider the complex blow-up of the diagonal $\triangle\subset Sym^2(\mathbb{C}^n)$. It is known that the blown-up space is smooth when $n=1,2$. I was wondering if that is still smooth for $n\geq3$? ...
3
votes
0answers
67 views

Set-theoretic generation by circuit polynomials

Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The algebraic matroid of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ ...
1
vote
0answers
87 views

Motives under de-singularization

Let $X$ be a singular variety over a field $k$ of characteristic 0. Suppose a minimal resolution of singularities of $X$, $f:\tilde{X} \rightarrow X$, exists, i.e., $\tilde{X}$ is a smooth projective ...
5
votes
0answers
85 views

Does the $K^1$-group of a complete flag variety vanish?

For $U(n)$ the Lie group of $n \times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space $$ U(n)/T^n $$ is called the complete flag variety of order $n$. For the special ...
2
votes
1answer
123 views

How to find a rational $\mathbb{F}_{\!q}$-curve on a quite classical Calabi–Yau threefold?

Take a finite field $\mathbb{F}_{\!q}$ such that $q \equiv 1 \pmod 3$, i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_{\!q}$, $\omega \neq 1$. Also, for $i \in \{0,1,2\}$ consider the elliptic ...
10
votes
1answer
748 views

Is there an algebraic version of Darboux's theorem?

Let $M$ denote a smooth manifold, and $\omega \in \Omega^2(M, \mathbb{R})$ a symplectic form. The classical version of Darboux's theorem states that for any $x \in M$, there exists an open ...
1
vote
0answers
104 views

Properness of algebraic stacks

What is the definition of a proper algebraic Artin stack? Is there a valuative criterion? If there is such a notion is it true that every fiber of a representable smooth morphism between two proper ...
2
votes
0answers
98 views

Characterization of effective descent morphism

A faithfully flat morphism of commutative rings $A \rightarrow B$ is an effective descent morphism. So is a regular monomorphism (right?). What is a characterization of effective descent morphisms? ...
1
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0answers
86 views

Is there a source in which Demazure's function $p$ defined in SGA3, exp. XXI, is calculated?

Suppose that $\mathcal{R}=(M,R,M^*,R^*)$ is a root datum. In section 1.2 of SGA3, exp. XXI, Demazure defines the $\mathbb Z$-linear map $p:M\to M^*$ by $$p(x)=\sum_{u\in R^*}(u,x)u$$ and proves many ...
1
vote
0answers
53 views

Relative normalization of a morphism of algebraic stacks

Let $S$ be a scheme and $f : \mathscr{X} \to \mathscr{Y}$ be a representable $1$-morphism of Artin stacks over $S$. Then what is the normalization of $\mathscr{Y}$ in $\mathscr{X}$? This is used in ...
3
votes
1answer
125 views

When does cohomology of a pro-algebraic group commute with filtered colimits of coefficients?

Let $G$ be a pro-algebraic group, that is, a projective limit of algebraic groups $G_i$. Let $V_i$ be an inductive system of finite dimensional rational $G_i$-representations, so that the colimit $V$ ...
2
votes
0answers
162 views

Construction of $K(Gal(\bar{k}/k), 1)$

Take any field $k$. Is there a construction of the Eilenberg-MacLane space $K(Gal(\bar{k}/k), 1)$ as a CW complex in terms of $k$?
1
vote
0answers
97 views

Is the restriction of a projection map open?

Let $X$ be a locally closed algebraic sub variety of $\mathbb C^{m+n}$. Let $p: \mathbb C^{m+n}\to \mathbb C^{m}$ be the projection map. Suppose that the restriction of $p$ to $X$ is injective. Is the ...
4
votes
1answer
111 views

Extending rational maps of nodal curves

Let $R$ be a discrete valuation ring with fraction field $K$ and $C, D$ two nodal (=prestable) curves over $\operatorname{Spec} R$. If I have a map $C_K \to D_K$ between the restriction of the curves ...
2
votes
1answer
143 views

Example of an intersection complex not concentrated in a single degree

I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful. I want to construct an example of an intersection ...
3
votes
0answers
166 views

Do Poincaré residue and integrable log connection commute?

Here are some basic notations and definitions: (ignore this part if familiar) 1.Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing ...
3
votes
0answers
104 views

Commutative group stacks and Galois cohomology

"Classically", if we consider an abelian variety $A$ over some number field $k$, we get a $Gal(\bar{k}/k)$-module $A(\bar{k})$, or equivalently a sheaf of abelian groups on the étale site $\...
7
votes
1answer
350 views

Hodge numbers rule out good reduction

A theorem of Fontaine says that if a geometrically connected smooth proper variety $X$ over $\mathbb{Q}$ has good reduction everywhere then $h^{i, j}(X)=0$ for $i\neq j$, $i+j\leq 3$. This means that ...
2
votes
0answers
195 views

Modern example of a reciprocity law and intuition behind it

I'm very new to the Langlands program and I was going through the Gauss reciprocity law, Hilbert's 9th problem, Artin's reciprocity law which allowed him to identify the Artin's L-functions with the ...
3
votes
0answers
99 views

Infinite sum in power series ring

Let $R$ be a commutative ring with $1$, $R[[x]]$ be the power series ring over $R$ and $A$ be an (prime) ideal of $R[[x]]$ with $x\not\in A$ and $\{f_i\}_{i=1}^\infty$ be a sequence of element of $A$. ...
2
votes
0answers
104 views

Motivic cohomology and resolution of singularities

Suppose $X$ is a singular variety and we somehow know its motivic cohomology groups. Can the knowledge of motivic cohomology help in understanding the singularities of $X$ and how to resolve them?
1
vote
0answers
46 views

Equivalent characterization of ordinary $F$-crystals

Let $k$ be a perfect field in characteristic $p$, let $W$ be its ring of Witt vectors. Let $A=W[[t_1,\cdots,t_n]]$, let $A_0=A/pA$. Let $H$ be an $F$-crystal over $A_0$. (An $F$-crystal over $A_0$ is ...
6
votes
2answers
320 views

Smooth projective surface with geometrically integral reduction

Let $S$ be a geometrically connected smooth projective surface over $\mathbb{Q}_p$. Can it be put in a proper flat $\mathbb{Z}_p$-scheme with a geometrically integral special fiber?
5
votes
2answers
277 views

Size of the category of cohomology theories

I'd like to understand the structure of the functor category Coh whose objects are cohomology functors from a category of Spaces to the category of graded commutative rings GCR. Spaces could be any of ...
0
votes
1answer
77 views

An example of a special $1$-dimensional non-Noetherian valuation domain

I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequence $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\...
2
votes
1answer
132 views

The locus of lines intersecting with another fixed line on a Fano threefold

Let $Y$ be an index $2$, degree $5$, Picard number $1$ Fano threefold, i.e $Y$ is a linear section of Grassmannian $\operatorname{Gr}(2,5)$. Let $\Sigma(Y)$ be the Hilbert scheme of lines on $Y$, it ...
4
votes
1answer
146 views

Proper morphisms with geometrically reduced and connected fibers

Let $f: X \to S$ be a proper morphism ($S$ locally noetherian), and $X \to S' \to S$ its Stein factorisation. By Zariski's Main Theorem the number of geometric connected components of the fibers of $f$...
2
votes
1answer
94 views

Lifting property for proper morphism

Let $X \subseteq \mathbb{C}^n$ be a complex affine variety and $\tilde{X} \to X$ a surjective proper morphism where $\tilde{X}$ is smooth. Is it true that every morphism $\mathbb{C} \to X$ can be ...
1
vote
0answers
114 views

Application of Zariski's Lemma other than Hilbert's Nullstellensatz

Zariski's Lemma is the following: Let $K$ be a field and $R$ be a $K$-algebra with $R=K[x_1,\dots,x_n]$ for some $x_1,\dots,x_n\in R$. If $R$ is a field then $x_1,\dots,x_n$ are algebraic over $K$. ...
3
votes
0answers
106 views

Borel Moore homology of classifying stacks

Let $G$ be a complex Lie group and $K\subseteq G$ its maximal compact subgroup. Is it true that $$H^\text{BM}_*(BG)\ =\ H^\text{BM}_*(BK),$$ and if not what is the relationship between them (e.g. what ...
4
votes
1answer
184 views

Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$

I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3) and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON $\...
8
votes
2answers
279 views

Fundamental group of a compact branched cover

My problem originates from the following classical result, proved, as far as I know, by Grauert and Remmert: Theorem. Let $Y$ be a compact complex manifold, $B \subset Y$ be a connected submanifold ...
2
votes
0answers
70 views

F-vectors of simplicial complex and f-vectors of non-faces of simplicial complex

Is there any result which gives us a relation between f-vector of simplicial complex and f-vector of nonfaces of a simplicial complex? Thank you
2
votes
1answer
279 views

In $\mathbb{C}[x,y]$: If $\langle u,v \rangle$ is a maximal ideal, then $\langle u-\lambda,v-\mu \rangle$ is a maximal ideal?

I have asked the following question at MSE and got one answer. Any further ideas are welcome: Let $u=u(x,y), v=v(x,y) \in \mathbb{C}[x,y]$, with $\deg(u) \geq 2$ and $\deg(v) \geq 2$. Let $\lambda, \...
1
vote
1answer
133 views

conditions on a morphism $f:X\rightarrow Y$ to ensure $X$ is reduced, given $Y$ is reduced?

Let $X,Y$ be finite type projective schemes over $\mathbb{C}$, and $f:X\rightarrow Y$ be a surjective morphism (but not an isomorphism). Suppose it is known that $Y$ is reduced, and the fibers of $f$ ...