# 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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27 views

### Is $h^1(X,O_X)$ always equal to the dimension of the Albanese?

Let $X$ be a projective integral scheme over $\mathbb{C}$.
If $X$ is smooth, then $\mathrm{h}^1(X,\mathcal{O}_X)$ is the dimension of the Albanese variety of $X$. Probably, even if $X$ is normal, ...

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43 views

### Toric Fan for the Du Val's singularities D_n and E_n

Let us consider the Du Val's singularities.
i.e. https://en.wikipedia.org/wiki/Du_Val_singularity.
It is well known that they are classified by ADE, because the exceptional divisors arising in the ...

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49 views

### Good reduction of rational surfaces

Let us work over $K = \mathbf{C}((t))$ for simplicity. We say that a smooth proper scheme $X/K$ has good reduction if it extends to a smooth and proper algebraic space $\mathcal{X}/\mathcal{O}_K$ ...

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86 views

### Hodge variation

I am reading Milne's online book of Shimura Varieties https://www.jmilne.org/math/xnotes/svi.pdf, I confused by a Definition of Hodge variation. On page 29, it was said something is called Hodge ...

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88 views

### Stacks in moduli spaces of sheaves research

I want to get some practice and build more appreciation for the use of stacks in the context of classical moduli spaces of sheaves. Here by classical I vaguely mean hands-on description of the ...

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35 views

### Non-degeneracy of the restriction of Poincaré pairing

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Then Poincaré duality shows the natural pairing $\left<-,-\right>$ on $...

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85 views

### Question regarding intersection product in Chow group of $\mathbb{P}^n\times\mathbb{P}^m $

Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{...

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90 views

### Are torsion-free rank 1 modules over integral schemes line bundles?

How far away are torsion-free rank 1 sheaves from the line bundles? Is there any condition that makes sure they are same? (for dimensions higher than 1). It is known that for a regular scheme of ...

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35 views

### What is the precise definition of a quadratic form of Minkowski type (in the infinite case)?

I've been trying to understand a construction in the paper "Degree Growth of Meromorphic Surface Maps" by Bouksom, Favre and Jonsson. In it they state,
In fact, the completion can be characterized ...

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53 views

### Linear independence of functions

Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...

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44 views

### Linearly independent functions evaluated at random points create full rank matrices

Assume $f_1, f_2,...,f_n: \mathbb{R}^d\mapsto\mathbb{R}^d$ are linearly independent functions. Now let $w_1,w_2,..,w_k\in\mathbb{R}^d$ be i.i.d. Gaussian random vectors distributed as $\mathcal{N}(0,\...

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**1**answer

101 views

### Blowing up vector bundles in the zero section

Assume we are given a scheme $X$ (feel free to add all the needed hypotheses, at this point I’m working with smooth schemes, but the fewer is needed, the better) and a vector bundle $E$ over $X$. I ...

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98 views

### Cohomology and base change theorem for non-noetherian schemes

Let $Y$ be a locally noetherian scheme, $f : X \to Y$ proper morphism, $\mathscr{F}$ a coherent module on $X$ which is flat over $Y$.
Then we have many theorems about the cohomology of $\mathscr{F}$ ...

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78 views

### Probability distributions on algebraic varieties [on hold]

Is there a notion of (algebraic) probability distribution on algebraic varieties? If there is, where can I find it? If not, why not?

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33 views

### Find an equation for a random set of points [on hold]

Let's say there is a random set of points, say 100, scattered over a 2d Cartesian plane, where no x-coordinate have more than one y-coordinate. Is there a way to find an(only 1) equation that will ...

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52 views

### Numerical and rational equivalences on intersection of divisors

Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...

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**1**answer

166 views

### Abel-Ruffini theorem for systems of polynomial equations

I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for ...

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**1**answer

79 views

### Combinatorial curves in combinatorial projective planes

Suppose $\mathcal{P}$ is an axiomatic projective plane (a nondegenerate point-line incidence geometry in which there is a unique line on any two different points, and a unique intersection point for ...

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95 views

### Hyperplane sections of non-singular projective toric varieties

Let $X^n\subset \mathbb{P}^N$ be an embedding of a non-singular projective toric variety (where variety stands
for a reduced irreducible scheme over $\mathbb{C}$, and toric means normal variety ...

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**2**answers

134 views

### Constant row-column sum matrices?

Given an integer $T$ how many $n\times n$ matrices in $\mathbb Z_{\geq0}^{n\times n}$ we have such that every row and column sum to same $T$?
Do the set of constant row and column sum matrices form ...

**3**

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**1**answer

145 views

### Motivic class of mixed Tate motive

Let $k$ be a field (of characteristic zero), $R$ be a ring and let $X\in DM(k;R)$ be a Tate motive. By definition, this means that $X$ is a summand of an object of the smallest strictly full ...

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**1**answer

47 views

### Relation between the Decomposition Invariants of a projective reduced Curve and its Normalization

Let $X$ be a reduced projective scheme over $k$ which is of pure
dimension 1. Let $\pi: X \to \mathbb{P}_k^1$ be a finite (hence
affine, surjective and flat) morphism of schemes having degree
$n$. ...

**3**

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**1**answer

62 views

### Connected sum of algebraic curves, handlebody decomposition, and induction on genus

Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...

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355 views

### Proj construction in derived algebraic geometry

The question
My question is easy to state:
Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”?
Given the vagueness of the question, you’...

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71 views

### Non-reduced base locus

I am looking for examples of ample line bundles over (possibly smooth) projective varieties whose base locus is either non-reduced or generically non-reduced.
The only example I can come up with is a ...

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**1**answer

121 views

### Question regarding Chow group of a blow-up

Let $X$ be a smooth complex projective variety, and $Y\hookrightarrow X$ be a smooth projective subvariety. Let $\pi:\tilde{X}\rightarrow X$ be the blow-up along $Y$, and let $j:E\hookrightarrow \...

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192 views

### Exact sequence in example in Grothendieck's Tohoku paper resulting from the Cech-to-derived-functor spectral sequence

Grothendieck gives in his Tohoku paper in example 3.8.3 an example for that $\check{\mathrm{H}}^{2}(X,\mathcal{F}) \neq \mathrm{H}^{2}(X,\mathcal{F})$.
In the beginning he states that there exisits ...

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**1**answer

87 views

### Morphisms from projective space to lower dimension spaces [duplicate]

Let $X$ be a variety over a base field $k$ of dimension $n$. Can there be non constant morphisms $P^m \to X$?

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746 views

### Link of a singularity

I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$.
If we set $x = x_1+ix_2, y = y_1+iy_2, z ...

**2**

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**1**answer

88 views

### Uniqueness of presentation for semi-abelian varieties

Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence
$$ 1 \to T \to G \to A \to 1$$
of algebraic groups, where $T$ is an algebraic ...

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181 views

### Trace of Frobenius on $p$-adic Tate module

Let $k$ be a finite field of characteristic $p>0$ with $q$ elements. Denote $K=W(k)[1/p]$.
Let $E$ be an elliptic curve over $W(k)$ with good reduction.
Choose a lifting $\mathrm{Frob} \in \...

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106 views

### Annihilator of an element and Jacobson radical

Let $R$ be a commutative ring with 1. Is there any characterization for an element $a$ of $R$ such that $\operatorname{ann}(1-a)\subseteq J(R)$ and $a\in J(R)$, where $\operatorname{ann}(x):=\{r\in R\...

**4**

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**1**answer

196 views

### Is a universally closed monomorphism a closed immersion?

The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-...

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95 views

### Exact sequence involving spectral data for Higgs bundles

In Beauville, Narasimhan, Ramanan's Spectral curves and the generalized theta divisor, Remark 3.7, the following exact sequence is presented:
$0 \rightarrow M(-\Delta) \rightarrow \pi^* E \...

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161 views

### Few questions about the algebraic cycles and the conjectures of Beilinson and Tate

I have three slightly related questions about algebraic cycles which I am just going to list them. I'd really appreciate any answers:
1) Is there any example of a smooth projective variety $X$ over a ...

**2**

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**1**answer

142 views

### Glueing modules over $\{x\}\times \operatorname{Spec} R$

Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, ...

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82 views

### Regarding local complete intersection morphism

Following Qing Liu's book Algebraic Geometry and Arithmetic Curves, section 6.3.2, we say that a morphism $f: X\rightarrow Y$ of varieties over some field $k$ is local complete intersection at $x\in X$...

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76 views

### Independent conditions imposed by a collection of double points

Let's consider the following statement : There exist a collection of $d$ points $\gamma \subset \mathbb{P}^{n}$, so that $h_{\Bbb P^n}(\gamma^{2} ,m) = \min\{(n+1)d, \binom {n+m}{n}\}$ implies for any ...

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188 views

### Is the schematic Fargues--Fontaine curve universally closed over $\mathbb{Q}_p$? [on hold]

I have been thinking about this question and it appears that the so-called Fargues--Fontaine curve may be relevant.
A definition is given in this document
A curve is a regular, Noetherian, ...

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**1**answer

269 views

### Basic example of a formal affine scheme, functorial point of view

$\let\opn=\operatorname$For my BA thesis I have to describe formal groups from the functorial point of view. I am hence reading Strickland - Formal Schemes and Formal Groups, which is apparently the ...

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149 views

### Coproduct in the category of affine schemes, functorial point of view

$\let\opn=\operatorname$An affine scheme is defined as a covariant representable functor $X:\opn{CRing} \to \opn{Set}$. The Yoneda embedding implies that the category of affine schemes, $\opn{...

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**1**answer

125 views

### Factorize a morphism into a morphism locally of finite type and a quasi-compact morphism

Does there exist a scheme not admitting a morphism locally of finite type to a quasi-compact scheme?
The reason I am asking this is that being locally of finite type and being quasi-compact are ...

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118 views

### Universal closure of schemes à la Nagata

Nagata compactification theorem is the following fundamental result:
Let $S$ be a qcqs scheme. Let $X$ be a separated $S$-scheme of finite type. Then there exists a proper $S$-scheme $\overline{X}$ ...

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158 views

### Glue DVR to itself, get a separated non-affine scheme

Here is what seems to be a fun little exercise in algebraic geometry. Take a DVR $R$ and automorphism of $Frac(R)$; glue $Spec(R)$ to itself via this automorphism. Can the glued scheme be separated ...

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59 views

### Defining pull-back of Chow groups under a morphism of special type

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety.
Let $\pi : Y\rightarrow X$...

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135 views

### Brauer Group of a nodal curve

What is known about the Brauer Group of a Nodal curve (complete integral curve) over $k$ with singularity as ordinary double point?
Is it trivial if $k$ algebraically closed?

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257 views

### Morphisms of smooth varieties

Let $f:X\rightarrow Y$ be a surjective morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that $Y$ is smooth projective and all the fibers are ...

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**1**answer

138 views

### Localization and containment in commutative ring

Let $R$ be a commutative ring with identity and $x, y $ be fixed elements of $R$ such that for each maximal ideal $m$ of $R$ we have $\langle \frac{x}{1_m}\rangle\subseteq\langle \frac{y}{1_m}\rangle$ ...

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111 views

### Exact sequence of normal cones

Suppose that we have a sequence $i: X \hookrightarrow Y$, $j: Y \hookrightarrow Z$ of closed embeddings of varieties such that $i$ is regular. In this case, do we have an exact sequence of cones of ...

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268 views

### A purely algebraic argument for existence of a section of a smooth projective morphism to the projective line

If I am reading this post correctly, any smooth projective $\mathbb{C}$-morphism of schemes $X\rightarrow \mathbb{P}^1$ admits a section. I am afraid of the topological argument presented there. Is ...