# 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

**4**

votes

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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**

vote

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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 $f$ ...