# 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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### Locus of trivialization of an extension of a vector bundle

Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...

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104 views

### Dimension of Prym variety of cover

I am reading the article by Lawrence and Venkatesh on diophantine problems and $p-$adic period mappings. At page $35$ they say that the dimension of the Prym variety of an (unramified) cover of curves ...

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95 views

### Rational curves on ruled surfaces

Let $S$ be a ruled surface (over an algebraically closed field) with an $\mathbb{P}^1$-bundle $\pi\!: S \to E$ onto an elliptic curve $E$. What is the classification of (possibly singular) irreducible ...

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87 views

### Elliptic curve model of the finite field extension

Recently Thorsten Kleinjung and Benjamin Wesolowsky published https://arxiv.org/abs/1906.10668
There is the definition elliptic curve model of the finite field extension:
Definition 2.1 (Elliptic ...

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88 views

### Complex solutions of a system of polynomial equations

Suppose we have a system of $n$ polynomial equations over $\mathbb{C}$ in $n$ unknowns and we know that it has more solutions in $\mathbb{C}^n$ than its Bezout's bound and, consequently, that it has ...

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246 views

### Representing $j_*\mathcal{O}_U$ as filtered colimit of perfect complexes

Let $X$ be a quasi-compact and quasi-separated scheme, and $U\subseteq X$ be a quasi-compact open subscheme. Then we can consider $Rj_*\mathcal{O}_U$ the (derived) pushforward of the structure sheaf ...

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203 views

### Interlacing sequences by polynomials?

Given $t=2^\ell$ where $\ell\in\mathbb N_{>0}$ and $M\in\mathbb Z$ and two sets of integers $\{a_1,\dots,a_t\}$ and $\{b_1,\dots,b_t\}$ with $0<a_1\leq \dots\leq a_t<M$ and $0<b_1\leq \...

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91 views

### Families over Artin Rings and Deformations

Let us work with a class of schemes over an algebraically closed field $k$ such that any two schemes in this class are isomorphic.
An example of such a class would be genus zero nonsingular curves ...

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79 views

### Is chern classes of holomorphic vector bundles enough to generate Hodge cycles [duplicate]

Let $X$ ba a smooth projective variety of dimension $n$. Hodge Conjecture states that every Hodge cycle in $Hdg^k(X,\mathbb{Q})$ comes from a Chern class of codimension $k$ in $CH^k(X,\mathbb{Q})$. ...

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53 views

### Non-minimal Krull associated primes of a PF-ring

A commutative ring $R$ is said to be a PF-ring if every principal ideal of $R$ is a flat $R$-module. Also, a prime ideal $P$ of $R$ to be a Krull associated prime of $R$ if
for every element $x\in P$ ,...

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251 views

### Simplicial resolutions of varieties

Let $X$ be a projective variety (over $\mathbb{C}$). Is there a simplicial object $X_{\bullet}$ in the category of smooth projective varieties and a morphism of simplicial varieties $X_{\bullet} \...

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147 views

### Fibers of blow up in families

Let $T$ be a smooth curve over $\mathbb{C}$ and $p:\mathbb{P}^n \times T \to T$ the natural projection. Let $V \subset \mathbb{P}^n_T$ be a $T$-flat subscheme of codimension at least $2$ and $\pi: \...

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116 views

### Algebra of meromorphic functions on a Riemann surface

Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...

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91 views

### Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$

Let $k$ be a field and $A$ a noetherian local $k$-algebra.
Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$.
Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...

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54 views

### The underlying curve of a family of genus zero $n$ punctured curves

Let $X$ be a curve of genus zero over an algebraically closed field $k$ so that $X \cong \mathbb{P}_k^1$. Let $(C, s_1, \cdots, s_n)$ a $n$ punctured genus zero curve over $k$ where $s_i: k \to C$ are ...

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336 views

### Structure of the variety of $n$-tuples of $m \times m$ matrices with zero product

Consider the functor sending a commutative ring $R$ to $\{(A_1,\dots,A_n) \in ( M_m(R) )^n | A_1 \dots A_n =0 \}$ which defines a scheme over $\mathbb Z$, let $X$ be its base change to $\mathbb C$.
...

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239 views

### Fundamental group of formal punctured disc and punctured affine line

On a course that ended some time ago, I was handed the following problem:
Problem: Compute $\pi_1^{ét} (\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}, \overline x)$.
Hint: Find all finite ...

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75 views

### Singularity of Brill-Noether sub varieties of Picard varieties of smooth curves

Suppose $C$ is a smooth projective curve over complex numbers. The singularities of the theta divisor $\Theta$ in $Pic^{g-1}(C)$ is described in the literature. It is $W^{1}_{g-1}=\{l\in Pic^{g-1}(C): ...

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161 views

### (Higher) extensions of mixed Hodge structures

Mixed Hodge structure is introduced by Deligne and it's very useful for studying complex algebraic varieties. We know $\text{MHS}$, the category of mixed Hodge structures is an abelian category. Where ...

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174 views

### Hochschild-Serre for étale cohomology vs. Galois cohomology

Let X be an irreducible smooth projective variety of dimension d over a number field K. Let $\bar X$ denote its base change to an algebraic closure of K. Then, if Understand it right, the Hochschil-...

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123 views

### Is the Chow ring of determinantal varieties known?

The 1988' book "Determinantal varieties" by Bruns and Vetter describes the Chow group of determinantal varieties in Chapter 8, but as far as I understand this book it seems to me that relations ...

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98 views

### Quantum cup product and Dolbeault cohomology

Let $X$ be a smooth projective variety over $\mathbb{C}$. We consider the small quantum cup product $\star$ on the deRham cohomology ring $\displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$. ...

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229 views

### Is there an algorithm to compute a Belyi map for the Riemann surface?

Let $y^2=x^5-x-1$ be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at $\{0,...

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185 views

### Do negative indecomposable bundles on curves have sections?

Let $X$ be a smooth projective curve, and $E$ an indecomposable vector bundle on $X$ with $\mathrm{deg} E<0$. Is it true that $H^0(X,E)=0$?
This is true if $E$ is a line bundle, which means it is ...

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387 views

### Isomorphism of the $\ell$-adic Tate module of an elliptic curve with CM

Let $E$ be an elliptic curve over $K$ (totally real number field) with complex multiplication by the field $L$. Let $\psi$ be the Grössencharacter associated to $E$, assume that $\psi$ of type $(-r,0)$...

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277 views

### Definition of geometric monodromy

Consider a polynomial $f
\in \mathbb C[x_1,\dots ,x_n]$. An atypical value of $f$ is a complex number about which $f:\mathbb C^n\to \mathbb C$ is not a topological fiber bundle. Writing $\mathrm{Atyp}(...

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225 views

### Abel-Jacobi map over a field

Suppose $X$ is a smooth projective variety over a field $k\subset \mathbb{C}$. Let $CH^r(X)_{hom}$ be the Chow group of codimension $r$ cycles defined over $k$ and homologous to zero. The usual Abel-...

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78 views

### Formal character of local cohomology groups with support in Schubert cells

Let $k$ be a field of characteristic zero, $G$ a connected semi-simple algebraic group over $k$ and $B$ a fixed Borel subgroup of $G$ with maximal torus $T$. Also denote by $W$ the Weyl-group of $G$. ...

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315 views

### Abel-Jacobi map on symmetric product of genus 4 curve

Suppose $C$ is a genus $4$ smooth projective curve over complex numbers.
The Abel-Jacobi map from $Sym^4(C)$ to $Jac(C)$ is birational.
Is this a blow-up along a surface or a curve. Can one determine ...

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149 views

### Vanishing of Chow groups in high codimension

Let $X$ be a smooth affine variety of dimension $n>2$ over $\mathbb{C}$. From the examples I have seen (admittedly very little) it seems to me that these varieties don't have torsion classes a ...

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135 views

### Definition of an invariant differential of an elliptic curve

I am somewhat confused by the definition of the invariant differentials in J. Silverman's book The Arithmetic of Elliptic Curves.
Let $E$ be an elliptic curve with Weierstrass equation $F(x,y)=0$. ...

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340 views

### True on stalks, false on affine opens [closed]

In scheme theory, there are some properties that can be specified purely on the stalks of the structure sheaf but they "lift" to the properties of the values of structure sheaf on affine opens, e.g.
...

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59 views

### Sections of Normal Sheaf and Tangent Space of Hilbert Scheme [duplicate]

Let $X= \mathbb{P}^n$ the projective space (interpreted as scheme) and $Y \subset X$ a closed subspace.
The existence of a called Hilbert scheme $H_X(Y)$ is well known which parametrizes closed ...

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243 views

### Algebraic geometry “over the function field” of the base

This is vaguely similar to, but quite different from, this question.
In the above linked question the focus is on fields of large cardinality per se. Here we fix a base field $k$ (say algebraically ...

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224 views

### Interesting algebraic geometry over large fields?

Are there any interesting algebro-geometric phenomena that happen over large algebraically closed fields of characteristic 0 and do not happen over $\mathbb{C}$ ("large" means cardinality larger than ...

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116 views

### Algorithm telling when an affine curve is planar

I am sorry, I asked a misguided question here: Reference request: smooth affine curves are planar, here is my attempt at a better question.
Let $\mathfrak{p}$ be a prime ideal of $\mathbb{C}[x, y, z]$...

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108 views

### Reference request: smooth affine curves are planar

Let $X\rightarrow\mathrm{Spec}\:\mathbb{C}$ be an affine smooth morphism of relative dimension$\leq 1$. What is a reference for the fact that there exists a $\mathbb{C}$-locally closed immersion $X\...

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240 views

### Progress on Bondal–Orlov derived equivalence conjecture

In their 1995 paper, Bondal and Orlov posed the following conjecture:
If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...

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67 views

### (Very) effective bounds on Nullstellensatz proofs for injectivity

Given $m$ multilinear polynomials $f_1,f_2,\ldots,f_m \in \mathbb{F}_2[x_1,x_2,\ldots,x_n]$ of total degree at most $2$, I want to efficiently determine if the mapping $\vec{x} \to (f_1(\vec{x}),f_2(\...

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572 views

### Number of solutions mod p and Betti numbers

Suppose $X$ is proper flat scheme over Sepc$\mathbb{Z}$. If $p$ is a large prime, then by Weil conjectures one can recover the Betti numbers of $X$ from the size of $X(\mathbb{F}_{p^n})$ for all $n$. ...

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135 views

### Local heights in Vojta's conjecture

I am a complex geometer trying to parse Vojta's conjecture on rational points, and I have a very basic misunderstanding (I apologize if this is too easy for MO).
Let $X$ be a variety over a number ...

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209 views

### Singularities of curves that are moving

Let $k$ be an algebraically closed field, let $d\ge 2$ be an integer and let $f,g\in k[x,y,z]$ be two homogeneous polynomials of degree $d$ without common factor.
We want to know what are the ...

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57 views

### complexity of system of equations defining affine variety

Say you have an affine variety $X$ in $n$-dimensional affine space. (You can even assume we are over $\mathbb{C}$, but I believe the nature of my question is algebraic).
I want to bound from above ...

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157 views

### Restriction of an ideal sheaf to a hyperplane

Let $Z$ be a zero dimensional subscheme in $\mathbb{P}^3$ and $H$ be a hyperplane. Then we have an exact sequence
$0 \to I_Z \to I_Z(1) \to^f I_Z(1) \mid_H \to 0$. On the other hand, we another exact ...

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291 views

### Relation between the Spec and the Proj of a ring

I am reading Thaddeus' paper on GIT and flips (https://arxiv.org/pdf/alg-geom/9405004.pdf), and I am confused with a claim in the begining.
Let $R$ be a finitely generated integral algebra over an ...

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178 views

### A deformation of the second Hirzeburch surface $F_2$ over $\mathbb CP^1$

I would like to know if there exists a smooth complex projective $3$-fold $X$ that admits a fibration $\pi: X\to \mathbb CP^1$ such that all fibers are smooth, $\pi^{-1}(0)$ is the second Hirzebruch ...

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99 views

### On a canonical bundle formula on a Calabi-Yau type variety

Let $(X,B)$ be a Calabi-Yau pair, that is, $(X,B)$ is lc (or klt for simplicity) and $K_X+B \sim_\mathbb{Q} 0$. Given a fibration $f:X \to Y$, there is an induced generalised pair $(Y,B_Y+M_Y)$ with $...

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304 views

### Deligne's Mixed Hodge Theory

Deligne constructs Mixed Hodge Structures (MHS) on the cohomology, $H^{*}(X)$, of an algebraic variety $X$, in his papers Hodge II and Hodge III. Really the question below is rather vague, and is ...

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243 views

### Representability of Grassmannian functor by a scheme

I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...

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149 views

### Intuition for Fano visitors

I have a hopefully simple question regarding derived categories of projective varieties. A question of Bondal asks if $Y$ is a smooth projective variety, is there a smooth projective Fano variety $X$ ...