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Questions tagged [ag.algebraic-geometry]

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

7
votes
0answers
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
1answer
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
1answer
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
0answers
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
votes
0answers
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
votes
0answers
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
vote
0answers
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
votes
0answers
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
103 views

Intermediate Jacobian of abelian varieties

Is the intermediate Jacobian of an abelian variety again an abelian variety?
1
vote
0answers
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
1answer
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
0answers
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
4answers
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
votes
0answers
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
votes
0answers
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
0answers
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
vote
0answers
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
1answer
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
vote
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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, ...