# Questions tagged [ag.algebraic-geometry]

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

15,369
questions

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### Is the induced map between etale cohomology groups independent of the compactification?

Let $k$ be an algebraically closed field, and $X$ be a smooth variety. For any compactification $i: X \hookrightarrow Y$ (so $X$ is a dense open subset of $Y$), consider the induced map
$i_!: H^i_{et,...

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89 views

### Iitaka dimension of a $\mathbb{Q}$-Cartier Prime divisor

Let $X$ be a normal projective variety and $D$ a prime divisor such that $mD$ is Cartier for some integer $m>0$.
Suppose $H^1(X,\mathcal{O}_X)=0$ and $mD|_D\sim 0$.
My questions are the following:
...

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**1**answer

207 views

### Useful invariants of etale topoi not coming from the shape

Given a "nice" scheme over a number field or a finite field, etale cohomology (in part because of its functoriality) provides some powerful invariants of the underlying scheme. Sometimes non-abelian ...

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200 views

### Is there a flat proper morphism that is not finitely presented?

Is there an example of a flat proper morphism of schemes $X\rightarrow S$ whose fibers are geometrically connected, reduced and have dimension 1, but which is not itself finitely presented? What ...

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114 views

### On the Beilinson's conjecture regarding the proper flat integral models

I had a question which seems to be straightforward but I wasn't able to figure it out. In
page 13 of this paper a conjecture of beilison is mentioned that if $\mathcal{X}_{\mathbb{Z}}$ is a proper ...

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141 views

### Compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ for a semi-abelian scheme $A$

How can I compute $H^i(S,\underline{\text{Hom}}(A,\mathbb G_m))$ (where $A$ denotes a semi-abelian scheme over $S$, $\mathbb G_m$ denotes the multiplicative group over $S$ and $\underline{\text{Hom}}$ ...

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350 views

### Finiteness of $H_1 \backslash G / H_2$ and the geometry of the orbits

Let $G$ be a connected reductive group over an algebraically closed field $k$. By the Bruhat decomposition, $P \backslash G/P \cong W_P \backslash W / W_P$ is a finite set for any parabolic subgroup $...

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68 views

### Number of nodes of a complete intersection lie on a plane

Suppose $X$ is a general smooth hypersurface of degree $\ge 6$ and $Y$ be an irreducible hypersurface of degree $\ge 2$. Let $X \cap Y$ has at least $5$ nodes. Is it possible that $4$ nodes of $X \...

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**1**answer

318 views

### Motivation for Suslin’s Rigidity Conjecture

Suslin Rigidity conjecture states that motivic cohomology
$$
H_{\mathcal{M}}^1(\operatorname{Spec}(F),\mathbb{Q}(n))
$$
of the field $F$ coincides with motivic cohomology for the subfield of ...

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260 views

### On the product in the power series ring

Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...

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84 views

### Polynomial time decodable binary linear codes achieving $GV$ bound?

Are there explicit or random construction of linear codes that achieve the $GV$ bound with polynomial time decodable property with alphabet size $q=2$?
Tsfasman, Manin, Vladut beat the bound at ...

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136 views

### Fields such that there is a single surface into which every curve embeds

For which fields $k$ does there exist a proper morphism $S\rightarrow \mathrm{Spec}\:k$ of relative dimension $\leq 2$ such that for every geometrically connected smooth proper $C \rightarrow \mathrm{...

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104 views

### Clifford index and Clifford dimension

Let $C$ be a smooth projective curve. Let $A\in Pic(C)$. The Clifford index of $A$ is defined as
$$Cliff (A)= deg\,A-2(h^0(A)-1).$$
What does this actually measure.
Next the Clifford index of $C$ is ...

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194 views

### Weight monodromy conjecture for surfaces?

Related question: weight monodromy conjecture for curves?
In order to compute the local zeta functions of two classes of Shimura surfaces at primes of bad reduction, Rapoport and Zink proved the ...

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488 views

### Strongly abnormal schemes

Call a scheme $Y$ proper positive-dimensional over $\mathrm{Spec}\,\mathbb{C}$ abnormal if there exists an irreducible scheme $X$ affine of finite type over $\mathrm{Spec}\,\mathbb{C}$ and a $\mathbb{...

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186 views

### Is there function that can be expanded as infinite power series with bounded positive coefficients？

Is there a rational function $F$ which may be expanded as power series with coefficients of unperiodical positive integers in such a form:
$$F(x)=\sum_0^{\infty}a_i x^i,\qquad a_i\in \mathcal{N} \cup ...

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167 views

### Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group

In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said
"We say that
$\rho$ is crystalline/de Rham/Hodge–Tate if ...

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146 views

### Which complexes of coherent sheaves are dual to perfect ones?

Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex $D$ of coherent sheaves. Then (if I understand these matters and the answer to the question Characterization of ...

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298 views

### Comparison of complex and p-adic Hodge structure

Let $X$ be a smooth projective variety over a p-adic local field $K$, and let $\bar{K}$ be the algebraic closure of $K$. Fix an isomorphism $\sigma:\bar{K}\to\mathbb{C}$. Do $\sigma$ induces an (iso)...

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380 views

### Do coherent sheaves on rigid analytic spaces form an abelian category?

It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (...

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77 views

### Nodes of intersection of hypersurface and a hyperplane

It is known that an intersection of a general hyperpursuface of degree $\ge 5$ in $\mathbb{P}^3$ and a hyperplane can have at most $3$ nodes. My question is the following:
Can these $3$ points lie ...

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330 views

### The ring of global sections of a regular scheme

Let $X$ be a Noetherian regular scheme. Is $\mathcal{O}_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.

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160 views

### How non-projective can a complete variety be?

Given a positive integer $n$, can you give an example of a connected smooth proper $\mathbb{C}$-scheme such that any $n$ closed points are contained in a common affine open, but there is $n+1$ closed ...

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99 views

### Dense Stein subset in complex manifold

Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor.
...

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381 views

### Original reference for Adams Riemann-Roch theorem

Let $f\colon Y\to X$ be a proper morphism between smooth quasiprojective $k$-algebraic varieties. Denote by $\psi^j$ the $j$-th Adams operation on the Grothendieck group of vector bundles and $\theta^...

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79 views

### Varieties not injecting into liftable ones

I give you a finite field $F$. Can you give me an example of a geometrically connected smooth proper $F$-scheme $X$ such that there is no monomorphism (in the category of $F$-schemes) $X\rightarrow X'$...

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113 views

### A canonical complex computing etale cohomology

Crystalline cohomology can be computed as the hypercohomology of the de Rham-Witt complex.
If we want to compute the etale cohomology of the constant sheaves $\mathbb{Z}_l$ or $\mathbb{Q}_l$ (well, ...

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138 views

### Are there some relations between F-polynomials and theta functions?

F-polynomials are certain polynomials appears in the expansion formula of a cluster variable, see for example the formula (6.5) in cluster algebras IV. Theta functions in the paper correspond to ...

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334 views

### Rationally connected Kähler manifolds are projective

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:
https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf
She writes in this remark the following:
...

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524 views

### Italian-style algebraic geometry in homotopy theory?

In homotopy theory, stacks can be occasionally useful (i.e. in the chromatic story). I come from a differential geometry background, so when people say that algebraic geometry is useful in homotopy ...

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256 views

### Pointless, non-singular, absolutely irreducible affine plane curves over finite fields

We think the following is true:
For all sufficiently large primes $p$ and all natural $g \ge 1$, there
exists affine plane curve $f(x,y)=0$ over $\mathbb{F}_p$ which
is non-singular, absolutely ...

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102 views

### Smooth absolutely irreducible (?) genus 1 plane pointless curve over $\mathbb{F}_{13}$

We got a family of genus 1 plane curves that may violate a bound
in a paper.
Explicitly: Let $F(x,y)$ be the degree 39 polynomial with integer coefficients:
...

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184 views

### Bertini type theorem for very ample line bundle

Let $X$ be a normal, projective variety (can take $X$ to be a hypersurface in a projective space) of dimension at least $3$. Let $L$ be a very ample line bundle on $X$, hence base-point free. What can ...

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59 views

### How can I find the integral orthogonal group of a given symmetric positive definite form?

I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix:
$$M=\begin{bmatrix}
x_1 &...

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90 views

### Catenarity and epimorphisms of rings

Let $R$ be a commutative ring. The following are well-known:
If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary.
If $R$ is catenary and $S\subseteq R$ is ...

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214 views

### Birational Invariants of regular surfaces

Let $X,Y$ surfaces (so $2$-dimensional proper $k$-schemes) which are regular (so the stalks are regular) and birational and denote by $f: X \dashrightarrow Y$ the corresponding rational birational ...

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282 views

### Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$

Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of ...

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163 views

### Ample vector bundles and embeddings

If $X$ is a complete variety over a field, a line bundle $L$ is said to be very ample if there is a closed immersion from $X$ into a projective space, such that the pullback of $\mathcal{O}(1)$ is ...

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150 views

### Étale morphism over unirational/uniruled variety

Suppose we have an étale morphism between smooth quasi-projective (complex) varieties $X \rightarrow Y$ and assume that $Y$ is unirational. I am wondering whether we can somehow deduce that $X$ is ...

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119 views

### Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement

Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...

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370 views

### Reference on Grothendieck trace formula

I need to refer to the so-called Grothendieck trace formula, but after checking tens of Google pages, I still cannot find a proper reference on this topic. Could anyone tell me some good book/papers ...

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132 views

### Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.
See for ...

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107 views

### Maximal unramified quotient of $E[p]$ for the action of $G_{\mathbb{Q}_p}$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good and ordinary reduction at an odd prime $p$.
Suppose $E[p]$ denotes the $p$-torsion points of $E$ and $G_{\mathbb{Q}_p} := \text{Gal}(\...

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519 views

### Does this degree 12 genus 1 curve have only one point over infinitely many finite fields?

Let $F(x,y,z)$ be the degree 12 homogeneous polynomial:
$$x^{12} - x^9 y^3 + x^6 y^6 - x^3 y^9 + y^{12} - 4 x^9 z^3 + 3 x^6 y^3 z^3 - 2 x^3 y^6 z^3 + y^9 z^3 + 6 x^6 z^6 - 3 x^3 y^3 z^6 + y^6 z^6 - 4 ...

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452 views

### Galois representations associated to the algebraic cycles and transcendental cycles of K3 surfaces

Given a K3 surface $X$, the cup product defines a non-degenerate even unimodular structure on the lattice $H^2(X,\mathbb{Z})$. Inside this lattice we have the Neron-Severi group $\text{NS}(X)$, which ...

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243 views

### Severi Formula for Intersection Multiplicities

I say in advance that I am a novice in Intersection Theory, so forgive me if my question is trivial.
Let $X\subseteq\mathbb{P}^N$ be a smooth irreducible projective variety of dimension $n$ and $V, W\...

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112 views

### Geometric interpretation of j-invariants of supersingular elliptic curves

In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all ...

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123 views

### Ramification divisor with base change

Let's work over $\mathbb{C}$. Consider the following commutative diagram
\begin{array}{llllllllllll}
E_1& \xrightarrow{f} &E_2\\
\downarrow{\pi} &&\downarrow{\pi}\\
P_1 & \...

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153 views

### Infinite Order Automorphisms of Planar Polynomials

Let $R_n$ be the integral polynomial ring $\mathbb{Z}[x_1,x_2,...,x_n]$, let $A_n$ be the group of ring automorphisms $\mathrm{Aut}(R_n)$, and for $f\in R_n$ let $\mathrm{Aut}(f)=\{\alpha\in A_n\ |\ \...

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126 views

### Properties of rings of global functions of open subschemes

It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but ...