# 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

**7**

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**0**answers

124 views

### Invariants of etale topological type that are not homotopy invariants

Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy ...

**3**

votes

**1**answer

175 views

### Extend Group Action of $\mathbb{A}^1 /G$ to Projective Line

My question refers to an argument used in Torsten Ekedahl's paper: https://arxiv.org/abs/0903.3148
in Example ii) (page 8):
We consider a finite subgroup of affine transformation of $\mathbb{A}^1$. ...

**1**

vote

**1**answer

167 views

### Intersection Solutions of four nonlinear equations

I have four nonlinear equations I want to find the points of intersection of these equations, and I used the software Mathematica, unfortunately after many hours of waiting it does not give me any ...

**3**

votes

**0**answers

89 views

### Is the generalized Kummer threefold rational in characteristics 3?

Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$:
$$
\sigma(x_i,...

**5**

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**0**answers

165 views

### Cohomology groups on small fppf site and small etale site are not the same

Let $F$ be a quasi-coherent sheaf on a scheme $X$. Is there an example where cohomology groups of $F$ on small fppf site of $X$ and small etale site of $X$ are not isomorphic?

**2**

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**0**answers

94 views

### Is there a definition of an unpointed schematic homotopy type?

In the paper Champs Affines (http://www.math.univ-toulouse.fr/~btoen/chaff.pdf) Toen introduces pointed schematic homotopy types (SHTs) to solve Grothendieck's schematization problem (described in ...

**1**

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**0**answers

63 views

### Module of Kahler differentials for manifolds [duplicate]

Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...

**4**

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**0**answers

169 views

### Basic questions about crystals and Grothendieck connections

I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. ...

**6**

votes

**1**answer

187 views

### Existence of isomorphism mod every power of the maximal ideal

This problem is a continuation of Hensel lemma and rational points in complete noetherian local ring.
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
Assume $...

**4**

votes

**0**answers

168 views

### Carayol's “ramified Eichler-Shimura relation” and its applications

In his paper "Sur la mauvaise reduction des courbes de Shimura" from '86 H. Carayol shows the following congruence relation:
Let $M$ be the tower of Shimura curves over a totally real $F$, associated ...

**16**

votes

**3**answers

809 views

### Are there prominent examples of operads in schemes?

There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...

**5**

votes

**1**answer

106 views

### Number of connected components of degree 2 affine algebraic varieties

Suppose an algebraic variety $V$ is given as the solutions to $q$ polynomial equations of degree $\le k$ with real coefficients
$$p_1(x_1,\dots,x_m)=0,\dots,p_q(x_1,\dots,x_m)=0$$
for $x\in\mathbb R^m$...

**1**

vote

**0**answers

33 views

### Counting Zeros Under Unitary Action

Assume we have two polynomials $f_1$ and $f_2$ with Newton polytopes $A_1 , A_2 \in \mathbb{Z}^2$. Also suppose that coefficients of $f_1$ and $f_2$ are generic. Then we pick a unitary matrix $Q$ and ...

**2**

votes

**0**answers

141 views

### Grothendieck group of constructible sets

Let $K_0$ be the Grothendieck group of complex algebraic varieties. This is the group generated by all complex algebraic varieties, subject to the relations:
(i) $[X]=[Y]$ if $X,Y$ are isomorphic,
(...

**5**

votes

**0**answers

296 views

### Completion of a local ring of an arithmetic surface

Vaguely, an arithmetic surface is an algebraic curve with coefficients in some "ring of integers".
More formally it is an integral normal scheme $X$ of dimension 2 of finite type over a Dedekind ...

**3**

votes

**1**answer

91 views

### Summing complete intersections

Suppose we have polynomials $f_1,\dots,f_r\in k[X_1,\dots,X_N]$ defining a complete intersection in $\mathbb{A}^N$. I suspect that it is then true that $f_1(X)+f_1(Y),\dots,f_r(X)+f_r(Y)\in k[X_1,\...

**3**

votes

**0**answers

112 views

### Etale cohomology of projective spaces in the rigid analytic setting

Take $K$ a complete non-archimedean field (maybe algebraically closed, to simplify the question), and $\mathbb{P}_K^d$ the rigid projective space over $K$. Can we compute the étale cohomology with ...

**4**

votes

**0**answers

56 views

### Formality of fixed points in the equivariant localisation

Let $X$ be a complex algebraic variety equipped with an algebraic $\mathbf{C}^{\times}$-action.
The Borel construction gives a map $f: \mathrm{E}\mathbf{C}^{\times}\times^{\mathbf{C}^{\times}}X\to \...

**5**

votes

**0**answers

103 views

### Intermediate Jacobian of abelian varieties

Is the intermediate Jacobian of an abelian variety again an abelian variety?

**1**

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**0**answers

143 views

### Rank 2 vector bundle with trivial first chern class is self-dual

I saw a statement used in the paper that E of rank 2 with $c_1(E) = 0$ is self-dual. I was wondering, how does one prove this statement? If it makes a difference, let the underlying variety be ...

**9**

votes

**1**answer

275 views

### Is there a substitution that relates every Fermat curve to an elliptic curve?

I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level.
A Fermat Curve of degree $n$ is the set of solutions to $x^...

**1**

vote

**0**answers

60 views

### Equivariant map between cones is flat?

Assume I have a locally complete intersection scheme $X\rightarrow \mathrm{Spec}\:\mathbb{C}$ and and affine space $A^n_{\mathbb{C}}$ ($n>0$), both endowed with a $\mathbb{C}^*$-action (for the ...

**8**

votes

**4**answers

617 views

### Motivation for definition of Quotient stack

I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows.
Let $G$ be a Lie group action on a manifold $X$ (left action). ...

**2**

votes

**0**answers

104 views

### Non-complete intersection of low degree

Let $X=V(P_1,\dots, P_r)\subset \mathbb P^{n+1}_k$ be a closed algebraic subset defined by polynomials of degree resp. $d_1,\dots,d_r$ such that $d_1+\cdots+d_r<n+1$. Is there a way to compute the ...

**12**

votes

**2**answers

359 views

### Threefolds of general type with no holomorphic forms?

In relation to this question, I would like to ask for examples of (complex) threefolds of general type with no (nontrivial) holomorphic form.

**3**

votes

**1**answer

78 views

### The singularties of the dicriminant loci of the Lagrangian fibration

Let $X$ be a holomorphic symplectic variety of dimension $2n$ and $\pi: X \to \mathbb{P}^n$ be a Lagrangian fibration. It is known that $\pi$ is smooth outside of the discrimiant divisor $\Delta$. ...

**3**

votes

**1**answer

104 views

### Dimension of hermitian rank at most $k$ matrices over quaternions

In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...

**2**

votes

**0**answers

72 views

### Algebraic theta-functions of level $2$ on an elliptic curve

Consider an elliptic curve $E\!: y^2 = x(x-1)(x-\lambda)$, where $\lambda \in k \setminus \{0, 1\}$ for some field $k$ of $\mathrm{char}(k) \neq 2$. Also consider the ample divisor $D = 2P_0$, where $...

**10**

votes

**1**answer

302 views

### Elliptic curve over projective line with four points of multiplicative reduction

Consider the elliptic surface $E$ with affine equation
$$y^2 = x(x-1)(x-t^2)$$
over the base $\mathbf{P}^1$ with parameter $t$ (with complex scalar field). Then $E$ has four points of bad ...

**14**

votes

**1**answer

868 views

### GAGA for stacks

I am curious about stacky generalizations of the following GAGA theorem:
If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is ...

**1**

vote

**0**answers

65 views

### Degree of Varieties and Segre's Embedding

Let $X$ be an irreducible complex projective variety of dimension $p$ and consider two projective embeddings $\iota\colon X\subseteq \mathbb{P}^n$ and $\iota'\colon X\subseteq \mathbb{P}^{n'}$. Denote ...

**11**

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**0**answers

212 views

### 3-fold of general type homeomorphic to rational 3-fold

Is there a smooth (complex projective) 3-fold of general type which is homeomorphic (in the complex topology) to a rational $3$-fold?
I am aware of such examples in complex dimension $2$, for ...

**12**

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**0**answers

164 views

### Is there an odd degree unirational parametrization of a cubic threefold?

A cubic threefold is a smooth degree $3$ hypersurface in $\mathbb{P}^4$. Is there a cubic threefold $X$ over any field $k$ (possibly of positive characteristic) and an odd degree rational map $\mathbb{...

**9**

votes

**0**answers

167 views

### Batyrev's theorem in non-algebraic case

Let $X$ and $Y$ be two bimeromorphic closed Kaehler manifolds with trivial real $c_1$. Is it true that $b_n(X)=b_n(Y)$ for $n\geq 0$?

**1**

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117 views

### Behavior of regularity under base change

Let $(R,\mathfrak{m})$ be a noetherian local domain. Let $(A,\mathfrak{n})$ be a regular local ring contains $R$ such that $\mathfrak{n}\cap R=\mathfrak{m}$ and $A$ is essential finitely generated as ...

**3**

votes

**1**answer

109 views

### Definition of a vertical ideal sheaf and a vertical fractional ideal sheaf

I'm working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10.1090/jag/...

**1**

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**0**answers

104 views

### What happens to a variety after a change of variables?

Suppose I have an irreducible affine variety $X \subseteq \mathbb{A}^n_k$.
Let us denote $X = \{ x \in k^n : f_j(x) = 0 \ (1 \le j \le M) \}$. $k$ is an algebraically closed field. Let $a_i \in k$, $...

**20**

votes

**2**answers

1k views

### Complex analytic vs algebraic geometry

This is more of a philosophical or historical question, and I can be totally wrong in what I am about to write next.
It looks to me, that complex-analytic geometry has lost its relative positions ...

**2**

votes

**0**answers

87 views

### How to calculate tautological classes of some varieties?

I want to calculate tautological classes of some subvarieties sitting inside some bigger variety in the cohomology ring of that bigger variety. For example, suppose we have a chain of closed ...

**2**

votes

**1**answer

155 views

### On a special type of subring of $\mathbb C[x_0,…,x_{q-1}]$

Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...

**2**

votes

**1**answer

82 views

### Relate solutions to a polynomial system in complex numbers to solutions in a finite field

Suppose I have a system of polynomials which are homogeneous but of distinct degrees that I want to solve simultaneously:
$$F_1(z_1,\ldots,z_n)=\cdots=F_m(z_1,\ldots,z_n)=0.$$
Let $X(\mathbb F)$ ...

**5**

votes

**0**answers

151 views

### Fargues's Theorem for $Spa(C,C^+)$ (rather than $Spa(C,O_C)$

$\DeclareMathOperator\Spa{Spa}$Fargues's Theorem for $\Spa(C,O_C)$ states that the category of (mixed characteristic) shtukas with one paw at $x_C$ is equivalent to the category of Breuil-Kisin-...

**3**

votes

**0**answers

229 views

### Is the intersection of two function fields over finite fields again a function field?

I am interested in the question above. I know that the answer is NO if the base field is for instance $\mathbb{Q}$ (the intersection of $\mathbb{Q}(x^2)$ and $\mathbb{Q}((x-1)^2)$ is $\mathbb{Q}$ ...

**11**

votes

**1**answer

377 views

### Finiteness or infiniteness for Galois representations with unusual Hodge numbers

Say a representation $\operatorname{Gal}(\mathbb Q) \to GL_n(\overline{\mathbb Q}_\ell)$ has big monodromy if the Zariski closure of the image of $\operatorname{Gal}(\mathbb Q) $ contains $SO_n$ or $...

**20**

votes

**2**answers

745 views

### Where's the best place for an algebraic geometer to learn some algebraic number theory?

There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...

**1**

vote

**0**answers

109 views

### SL(2,R) invariant which are not SL(2,C) invariants

Consider four points, $\sigma_i$ i=1,2,3,4 on the line $\mathrm{Im}(z) = 0$ in the complex plane $\mathbb{C}$. Does it exist a rational function of these four points which is $\mathrm{SL}(2,\mathbb{R})...

**9**

votes

**1**answer

284 views

### Are there infinitely many real multiplication fields of abelian surfaces over $\mathbb Q$?

Do there exist infinitely many real quadratic fields $F$ such that there is an abelian surface $A$ over $\mathbb Q$ whose ring of endomorphisms, tensored with $\mathbb Q$, is $F$?
Do there exist ...

**7**

votes

**1**answer

211 views

### Remark 12.8.8 in Arinkin--Gaitsgory

I can not understand Remark 12.8.8 in the preprint "SINGULAR SUPPORT OF COHERENT SHEAVES AND THE GEOMETRIC LANGLANDS CONJECTURE". I am somewhat embarrased by the degree of my confusion, hopefully ...

**8**

votes

**1**answer

302 views

### Is there a conceptual reason why the notion of “quasicoherent sheaf” is independent of the choice of topology?

Let $X$ be a scheme and $\mathcal S$ a site which is a full subcategory of the category $Aff/X$ of affine schemes with a map to $X$. If I understand correctly, the category $QCoh^\mathcal S(X)$ of $\...

**2**

votes

**1**answer

133 views

### Relative weight lattice

Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I_G$ is the set of simple roots. Let $P$ be a standard parabolic subgroup, ...