# Questions tagged [ag.algebraic-geometry]

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

14,707 questions

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votes

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174 views

### How to produce low genus curves on abelian surfaces?

I would like to find "simple" complex algebraic curves (i.e. low genus) on a complex abelian surface $A$ (which are not just abelian subvarieties or translates of them). For example, a genus 2 curve, ...

**8**

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**0**answers

234 views

### Status of the Sarkisov program a la Corti

To a birational map $f$ between Mori fiber spaces, one can associate the "Sarkisov degree", a triple $(\mu,\lambda,e)$ which I will not define here. To factor $f$ into a sequence of elementary ...

**17**

votes

**1**answer

609 views

### Embedding abelian varieties into projective spaces of small dimension

Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into.
Is $d(A)$ uniform over all abelian varieties of a ...

**4**

votes

**0**answers

116 views

### Nomality of blow-ups of higher dimensional rational singularities

A theorem (Theorem 8.1) of Lipman Lipman1969 shows that the blow-up of a rational singularity on a normal surface is still normal. Does anyone know under what (extra) condition Lipman's results can be ...

**3**

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**0**answers

127 views

### A finitely generated ring as the limit of a category of finite rings?

I was reading the book of McLane about Categories. In Yoneda's lemma he shows that a functor $F$ evaluated in a object $O$ can be written as the set of natural transformations between the ...

**7**

votes

**1**answer

273 views

### Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?

I apologize if the question is bit trivial for mathoverflow, but I asked on stack exchange a while ago and haven't got any answer.
Let $k$ be a field of characteristic $p > 0$. Consider the ...

**12**

votes

**1**answer

330 views

### Some basic questions on crystalline cohomology

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$.
Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...

**4**

votes

**1**answer

174 views

### Motivation for using etale topology in representability of functors problems

I am reading a paper that proves the representability for certain functors whose domain is the category of superschemes. The paper claims that to prove representability of functors (or possibly just ...

**2**

votes

**1**answer

105 views

### On connectedness of intersection of subgroups

I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected ...

**0**

votes

**0**answers

53 views

### Quadrics over the univariate function field with discriminant of minimal degree

Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...

**17**

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**1**answer

508 views

### Is being of general type stable under generization

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.
Definition. An integral projective ...

**9**

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**0**answers

174 views

### Quantum Hamiltonian reduction and Quantum Airy structure

I have been reading Kontsevich and Soibelman's "Airy structures and symplectic geometry of topological recursion" (https://arxiv.org/pdf/1701.09137.pdf) and having trouble understanding their Section ...

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80 views

### How does this $\chi \in X^*(P)$ define a line bundle on $P \backslash G$, where $G$ is a semisimple linear algebraic group

Let $F$ be a number field, and $G$ be a semisimple linear algebraic group over $F$. We let $P_0 \subseteq G$ be a minimal $F$-rational parabolic subgroup.
Let $P$ be a standard (i.e. containing $P_0$...

**1**

vote

**1**answer

76 views

### Equalizer of local analytic isomorphisms

Let $a,b : V\to W$ be two morphisms of smooth complex analytic spaces.
Assume $a$ and $b$ are local analytic isomorphisms.
Does the equalizer $U$ of $a,b$ exist as a smooth complex analytic ...

**4**

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191 views

### Algebraic Geometry Over the Surreal and Surrcomplex Numbers

I was wondering whether or not there is some kind of theory of algebraic geometry over the field of Surreal and Surrcomplex numbers?

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95 views

### Resolution of Kleinian Singularities using Hilbert schemes of points

Apologies in advance for the naive and rather speculative question.
In this blog post by John Baez, he paints a (perhaps not original) picture of how one might expect that the minimal resolution of a ...

**4**

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**0**answers

116 views

### Does linear independence imply algebraic independence for partitioned homogeneous polynomials?

Define a partitioned homogeneous polynomial of degree $d$ to be a polynomial in $$\mathbb Z[x_{11},\dots,x_{1n},\dots,x_{d1},\dots,x_{dn}]$$ with monomials from entries in (polynomials that are $d$-...

**3**

votes

**1**answer

133 views

### Effective Cartier divisor is an open property

Let $X$ be a regular affine $\mathbb{C}$-scheme, $A$ a (finitely-generated) $\mathbb{C}$-algebra. Let $Y \subset X \times \mathrm{Spec}(A)$ be a closed subscheme of codimension $1$ such that for each $...

**1**

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139 views

### Continuity of Intersection Pairing on Chow monoids

Let $X$ be a smooth irreducible complex projective variety. As we know, if $\alpha,\beta$ are two cycles intersecting properly in $X$, we can define, via Serre's Intersection Formula, their ...

**1**

vote

**1**answer

122 views

### Image of smooth curve containing the image of a point as smooth point

Let $f: X \to Y$ be a finite, surjective morphism of smooth, quasi-projective varieties over a field $k$ of characteristic zero. Let $p\in X$.
If $\dim X>0$, then does there necessarily exist a ...

**2**

votes

**1**answer

91 views

### is the induced map of an embedding an Iso on Ext-groups?

I am sorry, but I am quite new to Ext groups of sheaves. However, I have a closed embedding of projective $\mathbb{C}$-schemes $\iota : X \hookrightarrow Y$ and was wondering if
$$\iota_*:\mathrm{Ext}^...

**2**

votes

**1**answer

102 views

### Action of birational map $f$ on the divisor class of line $[H]$

In the paper "Normal Subgroups in the Cremona Group", it is stated that the induced isometry $f_{\ast}$ of $f\in J_d$, where $J_d$ denote the set of Jonquières transformations of degree $d$, satisfies ...

**6**

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**1**answer

179 views

### smoothness of Hurwitz spaces with arbitrary ramification profiles

Fix integers $n\ge 3,d\ge 2$, and partitions $\lambda_1,\ldots,\lambda_n$ of $d$. Let $\mathcal{H}$ be the moduli space of degree $d$ covers $f:C\to\mathbb{P}^1$ that have ramification profiles $\...

**4**

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**1**answer

78 views

### Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$

Is it correct that any holomorphic Poisson structure on $C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $C^n$? By homogeneous I mean a quadratic Poisson structure of the form $\{z_i,...

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77 views

### Do Plucker relation follow from a subsystem of equations?

The following system of equation: $i, j \in \mathbb{Z}_{\ge 1}$, $i+1<j$, $J \subset \mathbb{Z}_{\ge 1}$, $J \cap \{i,j,i+1,j+1\} = \emptyset$,
\begin{align*}
P_{i,j,J}P_{i+1,j+1,J} = P_{i,j+1,J} ...

**2**

votes

**1**answer

210 views

### Is the support of a flat module generically flat?

Let $X$ be an affine, complex variety, $A$ be a $\mathbb{C}$-algebra (not necessarily noetherian) and $F_A$ is a coherent sheaf over $X \times \mbox{Spec}(A)$, flat over $\mbox{Spec}(A)$. Denote by $Y ...

**1**

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**1**answer

165 views

### Leray spectral sequence from hypercohomology

Context: Deligne, Theorie de Hodge II, section 1.4.8.
Let $f:X\rightarrow Y$ be a map between spaces; $\mathcal{F}$ a sheaf of abelian groups on $X$, and $\mathcal{F} \rightarrow \mathcal{F}^{\...

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172 views

### About quotient varieties

Let $K$ be a field, $L/K$ be a finite Galois extension with Galois group $G$ such that $(char(K),|G|)=1$ and $K$ contains all $|G|$th roots of unity. Let $B$ be a $L$-algebra of finite type endowed ...

**2**

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136 views

### Examples of endomorphisms of complex curves

I am looking for examples of smooth projective complex curves $X$ of genus at least $2$ and with algebraic endomorphisms $f:X\rightarrow X$ of degree at least $2$. In the case of elliptic curves and $\...

**4**

votes

**1**answer

125 views

### $h$ is identity as soon as $h(\Sigma)\cap \Sigma$ contains at least 5 points

In the paper "Normal Subgroups in the Cremona Group", under remark 5.1 they stated that for any generic set $\Sigma \subset \mathbb{P}^2_\mathbb{C}$ of $k$ points, and $h$ is an automorphism of $\...

**6**

votes

**1**answer

268 views

### How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...

**9**

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246 views

### Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?

$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...

**7**

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114 views

### Are maximal tori conjugate Zariski-locally?

Let $S$ be a scheme and let $G\to S$ be a reductive group scheme. Then $G$ admits a maximal torus etale-locally, and any two maximal tori are conjugate etale-locally, by Theorem 3.2.6 and Corollary 3....

**3**

votes

**1**answer

221 views

### Restriction of Ext sheaves on closed subschemes

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...

**4**

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89 views

### A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$

Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...

**2**

votes

**1**answer

264 views

### Can one always find a bigger global resolution

Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-...

**2**

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**0**answers

111 views

### Restriction of the sheaf of relative differentials

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials.
For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...

**3**

votes

**0**answers

73 views

### Isogeny to a semi-simple group

Let $G$ be a split reductive group over a field of characteristic zero. Is there always an isogeny $G \to H \times T$ with $H$ semi-simple and $T$ a split torus ?
I have in mind the case of $\text{GL}...

**10**

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205 views

### Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...

**2**

votes

**0**answers

130 views

### Structure of the End group scheme of an abelian scheme over ring of integers

Let $O$ be the integer ring of a p-adic field $K$ (finite extension of $\mathbb Q_p$), $\mathscr{A}$ be an abelian scheme over $S=\operatorname{Spec O}$, consider the group endohomorphism scheme of $\...

**0**

votes

**1**answer

107 views

### About a theorem of Mestre

There is a theorem of Mestre witch states:
If $K$ is any field, let $p(x) \in k[x]$ is a monic polynomial with degree $2n$, then there exist polynomials $g(x)$ and $r(x)$ with:
1) $g(x)$ and $r(x)$ ...

**3**

votes

**0**answers

119 views

### Homotopy colimit description of stacks

Let $F$ be an Artin stack. If $p: X \to F$ is an atlas for $F$, can we express $F$, in the $\infty$-category ${\rm Shv}^{\acute{et}}(k)$ of higher stacks, as a homotopy colimit over the simplicial ...

**1**

vote

**0**answers

81 views

### Bounding the denominator in the canonical bundle formula

My question concerns with Theorem 3.1 in the paper "A canonical bundle formula" by Fujino and Mori.
The theorem claims the following:
Suppose $X \to C$ is a fiberation whose general fiber $F$ has ...

**6**

votes

**0**answers

126 views

### Moduli space of flat connections of Lie group over a 2-torus

We know that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$
Namely,
$$
M_{\rm flat}=\mathbb{E} / {S}_N = \mathbb{P}^{N-1}
$$
...

**0**

votes

**0**answers

113 views

### Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form
$$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$
$$\vdots$$
$$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$
where $h_1(x_1,\dots,x_{...

**10**

votes

**1**answer

397 views

### Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.
Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$.
Let $f : X\to Y$ be a morphism of ...

**3**

votes

**0**answers

127 views

### Maximum number of integral roots in degree $d$ polynomial?

Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...

**6**

votes

**2**answers

271 views

### Outer Hodge groups of rationally connected fibrations

I believe the following is true and well known.
Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $\mathbb{C}$. Let
$$
f\colon X\rightarrow Y
$$
be a surjective map with ...

**0**

votes

**0**answers

109 views

### Final step in Coppersmith?

In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...

**2**

votes

**1**answer

120 views

### The centralizer of a semisimple element which is not contained in any proper parabolic subgroups

Let $G$ be a connected, reductive group over a field $k$. Let $A_G$ be the split component of $G$. If necessary, assume $k$ is perfect. Let $g \in G(k)$ be a semisimple element. Then the ...