# 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

**4**

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108 views

### Lefschetz theorem for the fundamental group of quasi-projective varieties over $\mathbb C$

Let $𝑋$ be a smooth quasi-projective variety of dimension at least 2. The following generalization of Lefschetz Theorem is proved for instance by Hamm and Le dung Trang: if $H$ is a general ...

**1**

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89 views

### Replacing a non-basepoint free line bundle by a basepoint free one

In his paper “Brill-noether-petri without degenerations”, Lazarsfeld says the following in in Corollary 1.4.
Statement - Assume that every member of the linear series $|C_0|$ in a K3 surface is ...

**0**

votes

**1**answer

113 views

### Family of zeros of polynomials

Let $k$ be an infinite field and $P(X_1,\dots,X_n)\in k[t][X_1,\dots, X_{n}]$, suppose that there exists a finite field extension $L$ of $k$ such that
$P(x'_1,\dots,x'_n)\in L[t]^{*}=L^{*}$ with $x'...

**9**

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487 views

### Conjectures of Peter Scholze about q-de Rham complex: examples

Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper
Canonical $q$-deformations in arithmetic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–...

**7**

votes

**1**answer

323 views

### Confusion about good reduction

I am confused about the notion of good reduction. Let $R$ be a DVR, let $K$ be its fraction field. If we have a smooth proper $K$-scheme $V$, then I believe $V$ is said to have good reduction at the ...

**3**

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81 views

### A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$

Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper:
Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra ...

**5**

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**2**answers

126 views

### Linear systems separating points

Is it easy to find an example of a complete linear system on a smooth projective curve (say over $\mathbb C$) which separates points but which is not an embedding?
(for just a linear system, one ...

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86 views

### Lifting a linear surface from a curve to the ambient surface

Let $X$ be a complex K3 surface and $C$ a smooth curve on $X$ and $A$ a basepoint free line bundle on $C$.
Aprodu's paper - Lazarsfeld Mukai bundles and applications says this. We cannot lift the ...

**4**

votes

**1**answer

219 views

### How to construct cup-product in a general site?

Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows?
...

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89 views

### Base change and family of stable maps

Suppose that family F of stable maps given by maps $f:C \to S,\mu:C \to P^r$ and sections $\rho_i:S \to C$
Suppose that $\Sigma(F)$ be union of all one dimensional components of locus of nodes in ...

**7**

votes

**2**answers

271 views

### Rational functions on reduced complex varieties that extend to global holomorphic functions

Suppose $A$ is an integral domain and a finite type $\mathbb{C}$-algebra. Let $X := \text{Spec}(A)$ and $K := \text{Frac}(A)$ be the fraction field. Suppose $h \in K$ is a rational function that ...

**5**

votes

**1**answer

179 views

### Coarse moduli space versus Kuranishi family

We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M_h$ the coarse ...

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126 views

### Quotient of an affine scheme by an étale finite group

Let $G$ be a finite étale group scheme over a field $k$ and $X=\mathrm{Spec}(A)$ be an affine scheme on which $G$ acts. The categorical quotient $X/G$ exists and may be described as $\mathrm{Spec}(A^H)...

**2**

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66 views

### Homology of linear groups over integral domains and their field of fractions

Let $A$ be a noetherian integral domain of finite Krull dimension with the field of fractions of $F$. Consider the natural injections $i_n:GL_n(A)\hookrightarrow GL_{n+1}(A)$, $j_n:GL_n(F)\...

**2**

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150 views

### Comparison of weight-monodromy filtrations

Setup:
Let $R$ be a finitely generated subring of $\mathbb{C}$. Let $X \rightarrow \mathbb{A}^1_R$ be a proper morphism of $R$-varieties, smooth except over a rational point $s \in \mathbb{A}^1_R$ ...

**2**

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**1**answer

172 views

### When $\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)$? [closed]

Let $\phi:G\rightarrow H$ be a morphism of (linear or not?) algebraic groups. What are, in general, the conditions to assure
$$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$

**0**

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122 views

### Affine open with an irreducible complement

Let $X$ be an integral scheme such that the morphism to $\mathrm{Spec}(\mathbb{Z})$ is proper. Assume the morphism to $\mathrm{Spec}(\mathbb{Z})$ has well-defined relative dimension and that the ...

**1**

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**0**answers

82 views

### Log-canonical bundle of a smooth curve with marked points

I am not sure if this question is appropriate for this site, but here it goes. I am not a geometer, so I am not familiar with notation in the area.
I am interested in the moduli space of $r$-spin ...

**2**

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**0**answers

74 views

### Log deformations in obstructed case

I'm going to assume reader is aware of semi-stable log structures either in Kawamata-Namikawa version or later approaches.
Anyway, let $X$ be a d-semistable variety. I want to know whether I can ...

**6**

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**1**answer

266 views

### Flat with smooth fibers implies formally smooth?

Is every flat morphism of schemes $X\rightarrow S$ such that the fiber over any point is smooth necessarily formally smooth? There are formally smooth morphisms that are not flat so the converse fails....

**0**

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**1**answer

241 views

### Formally smooth with smooth fibers but not smooth

Is there a simple example of a morphism of schemes $X\rightarrow S$ that is
formally smooth
for any point $p\in S$, the base change $X\times_S\mathrm{Spec}\:k(p)\rightarrow \mathrm{Spec}\:k(p)$ is a ...

**3**

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**0**answers

124 views

### Closed subvariety that is unique in its small analytic neighborhood

Let $Y$ be some smooth projective variety over $\mathbb C$ with $\dim Y \geq 2$. For a closed sub-variety $X \hookrightarrow Y$, consider the following property:
There is some small open neighborhood ...

**2**

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**4**answers

692 views

### A complete formalization of EGA in Lean

I have been lately thinking about the feasibility of creating a "mediocre algebraic geometer" AI. I thought that to train it, one could feed it some large chunks of algebraic geometry presented in an ...

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82 views

### Equivariant sheafs and $G$ actions on modules

I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf.
The set up ...

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**1**answer

167 views

### Example of a nonsmoothable scheme

I try to understand Iarrobinos example of a nonsmoothable 0-dimensional scheme with the help of Artins notes on it:
http://www.math.tifr.res.in/~publ/ln/tifr54.pdf (pages 4-6)
But I have some ...

**2**

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**1**answer

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-...

**0**

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195 views

### Jouanolou's trick in the non-projective case

Given a connected scheme $X$ proper over $\mathbb{C}$, does there exist a scheme $X'$ affine and of finite type over $\mathbb{C}$, and a Zariski locally trivial $\mathbb{C}$-morphism $X'\rightarrow X$ ...

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142 views

### The complex points of an algebraic $\mathbb{C}$-scheme have the weak homotopy type of a finite CW complex

Can you prove the following
Let $X$ be a scheme of finite type over $\mathbb{C}$, then the weak homotopy type of the complex-analytic space $X(\mathbb{C})$ is the weak homotopy type of a finite CW ...

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354 views

### Why are algebraic schemes called algebraic?

In scheme theory, an algebraic scheme is the data of a scheme + a morphism of finite type to the spectrum of a field. Where does the term "algebraic scheme" come from? It does not seem intuitive to me ...

**5**

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**1**answer

221 views

### Projective subvarieties of a quasiprojective variety

Let $X$ be a quasiprojective variety over $\mathbf C$. Take the union of all projective subvarieties $W \subseteq X$ that have dimension at least $1$. Is the result Zariski closed?
(I was wondering ...

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156 views

### The dimension of open subschemes

Let $X$ be an integral scheme. Assume there exists a morphism of finite type from $X$ to the spectrum of a field. Is the Krull dimension of any non-empty open $U\subset X$ (possibly non-affine) equal ...

**5**

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295 views

### Over what fields does the Mordell conjecture (Faltings's theorem) hold?

Inspired by this question, over what fields is the Mordel conjecture known to be true?
For instance, is it true over fields of finite type (that is, fields finitely generated over their prime ...

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141 views

### Abelian varieties with rank 0 over each global field

For each global field $K$, can we always find an Abelian variety $A$ with $A(K)$ rank $0$?
By Lang-Neron, Mordell Weil is also true for finite type fields (fields finitely generated over their prime ...

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140 views

### The Grothendieck ring of varieties with classical Zariski

Consider $\mathcal{V}_k$, the category of $k$-varieties over the finite field $k = \mathbb{F}_q$ with $q$ elements. We see varieties in the old "classical" sense of the word, foreseen with the old ...

**3**

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114 views

### Lifting of curves in characteristic zero

Let $K$ be an algebraically closed field of characteristic zero. Let $G$ be an affine reductive group over $K$, and let $H$ be a closed reductive subgroup of $G$.
Since $H$ is reductive the GIT ...

**-1**

votes

**1**answer

105 views

### Counting points on a scheme of finite type over an infinite field

Let $k$ be an infinite field. Let $f:X\rightarrow \mathrm{Spec}\:k$ be a morphism of finite type. Assume that $X$ is not the empty scheme and that $f$ is not of relative dimension $\leq 0$ (definition)...

**4**

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110 views

### Weights of mixed sheaves

In Weil II, a mixed sheaf is defined as a sheaf admitting a finite filtration with punctually pure quotients, and its weights are defined as the weights of the graded pieces. Are the weights of a ...

**3**

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**0**answers

150 views

### Simple description of a Grothendieck topology on the opposite of f.p. complex algebras

Let ${\cal A}$ be the category of finitely presented $\mathbb{C}$-algebras.
Let $J$ be the largest subcanonical Grothendieck topology on ${{\cal A}^{op}}$ such that the local algebras in $\cal A$ are ...

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123 views

### Cokernel of section of a general coherent sheaf

Given a scheme $X$ and an $\mathcal{O}_{X}$-module $\mathscr{E}$, we know that a section $s \in H^{0}(X, \mathscr{E})$ is equivalent to a morphism $s :\mathcal{O}_{X} \to \mathscr{E}$. It is the ...

**3**

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73 views

### Differential criterion for regular sequences

Let $R$ be a subalgebra of the polynomial ring $\mathbb{C}[X_1,\ldots,X_n]$.
Suppose $\theta=(\theta_1,\ldots,\theta_p)$ is a sequence of of elements in $R$. If I want to test for their algebraic ...

**12**

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**1**answer

300 views

### Cotangent Complex in Analytic Category

I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold ...

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125 views

### A birational morphism of a finite cover to itself

Let $X$ and $Y$ be normal projective varieties.
Let $\pi:X\to Y$ be a finite surjective morphism and $\tau:X\to Y$ a birational morphism.
Question: will $\tau$ be isomorphic? or any counter-example?
...

**5**

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**1**answer

213 views

### Fibre of GIT morphism

Let $ V $ be an affine variety (over $ \mathbb C$) with an action of a reductive group $ G$. I would like to consider the morphism $$ \pi : V \rightarrow V // G = Spec \, \mathbb C[V]^G $$
Let $ v \...

**2**

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**0**answers

190 views

### All curves over an infinite field embed into the projective space

Let $k$ be an infinite field. Let $X$ be a separated scheme of finite type over $k$. Assume $X$ have relative dimension $\leq 1$. Does there exist a locally closed immersion $X\rightarrow \mathbb{P}^...

**5**

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131 views

### Henselian Schemes

I have a question about properties of Henselian ring/schemes exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":
Here the relevant excerpt:
In the excerpt we start with ...

**2**

votes

**1**answer

160 views

### Restriction of a Cartier Divisor

Let $X$ be a surface (so $2$-dimensional proper $k$-scheme)
$D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and
$C \subset X$ a ...

**4**

votes

**1**answer

92 views

### Quasi-compact quasi-separated induction?

I believe I've encountered the statement below, but I've lost my reference and am unable to find another one. So, I'm posting this question to see if someone can give a reference, or at least confirm ...

**10**

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183 views

### Holomorphic versus algebraic $\mathbb C^*$-actions

I believe that the following is true:
Statement. A holomorphic $\mathbb C^*$-action on a complex projective manifold is algebraic if and only if it has a fixed point.
Where can I find a proof of ...

**1**

vote

**2**answers

226 views

### Regularity of certain schemes

In a book I am reading, "Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents" by Luc Illusie, Yves Laszlo, Fabrice Orgogozo (https://arxiv.org/abs/1207....

**3**

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134 views

### About exact sequences of vector bundles

I am currently trying to understand "Proprietes de descente des varietes a fibre cotangent ample" by M. Martin-Deschamps, and I can't understand the first corollary. That is : given a scheme $S$ of ...