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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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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0answers
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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1answer
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 ...
5
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0answers
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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0answers
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 $...
2
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0answers
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 ...
3
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0answers
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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0answers
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 ...
2
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0answers
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 ...
2
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0answers
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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0answers
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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0answers
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 ...
3
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1answer
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 ...
3
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0answers
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 $\mathscr{F}$ ...
7
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1answer
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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0answers
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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2answers
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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1answer
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 ...
5
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1answer
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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0answers
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 ...
7
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1answer
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 ...
1
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0answers
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 $...
5
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1answer
250 views

Reductive groups in algebraic geometry [duplicate]

In a lot of fields in algebraic geometry (e.g. GIT or topics on étale cohomology) which make use of group scheme concepts (or in more tame way of algebraic groups), the class of reductive algebraic ...
1
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1answer
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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1answer
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 ...
5
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2answers
1k views

Number of irreducible and connected components constant in flat families

A) Let $f:F\rightarrow S$ be a flat proper morphism of schemes with geometrically normal fibers. Then supposedly the number of $\textbf{connected}$ components of the geometric fibers is constant. ...
2
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1answer
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 \...
2
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0answers
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 ...
4
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0answers
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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0answers
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 \...
11
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4answers
3k views

Complex torus, C^n/Λ versus (C*)^n

I'm having trouble distinguishing the various sorts of tori. One definition of torus is the algebraic torus. Groups like $SU(2,\mathbb{C})$ and $SU(3,\mathbb{C})$ have important subgroups that are ...
8
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3answers
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 ...
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0answers
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\...
2
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1answer
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 $P^m \to X$?
44
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0answers
2k views

Uniformization over finite fields?

The following is a question I've been asking people on and off for a few years, mostly out of idle curiosity, though I think it's pretty interesting. Since I've made more or less no progress, I ...
2
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1answer
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 ...
9
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4answers
4k views

Definition of étale for rings

Let $A \to B$ be a ring extension. What is the definition of $B/A$ étale ? When $A$ is a field, do we get a nice characterization ?
4
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1answer
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-...
2
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1answer
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, ...
4
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1answer
668 views

Generalized Behrend version for Grothendieck-Lefschetz trace formula

[MOVED HERE FROM MSE.] The statement of the Grothendieck-Lefschetz fixed point theorem is well-known. For a proper algebraic variety $X$ over $\mathbb F_q$, $$\#X(\mathbb F_q) =\sum_i (−1)^i Tr(Fr_X, ...
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0answers
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 ...
3
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0answers
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 \...
2
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0answers
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 ...
5
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2answers
681 views

Remove denominators in de Rham cohomology

Let $\omega = \mathrm d \eta$ be an exact rational $n$-form on $\Bbb P^n$. It may happen that the polar locus of $\eta$ is not included in the polar locus of $\omega$. But is it true that $\omega = \...
2
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0answers
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 $x\in X$...
3
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1answer
136 views

Deformation of stable curve with regular total space [duplicate]

Let $k$ be a field, let $X/k$ be a stable curve. Is it always possible to find a deformation $\mathcal{X}/k[[t]]$ such that $\mathcal{X}$ is regular? (Sorry for the confusion, this is a duplication ...
0
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1answer
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 ...
0
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0answers
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, ...
2
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1answer
170 views

Proving the representability of a functor that is covered by open subfunctors

I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-...
4
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1answer
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 ...