# Tagged Questions

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

**0**

votes

**0**answers

20 views

### line bundle descents?

Let the permutation group $S_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S_4$. It is a projective variety by a theorem of Mumford. ...

**11**

votes

**1**answer

450 views

### Constructive proof of “Projective implies proper”

For every ring $A$, the structural morphism of schemes $\pi_A : {\bf P}^n_{A} \to {\rm Spec}{A}$ is a closed map. The usual proof of this fact is not constructive : given equations of a closed subset ...

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vote

**0**answers

109 views

### Explicit examples of Hasse--Weil zeta-function calculations for curves

The problem of calculating Hasse--Weil zeta-function for a given curve $C/\mathbb{F}_p$ over a finite field is far from being easy, especially for large genus (as discussed by Wouter Castryck at ...

**1**

vote

**1**answer

191 views

### Divisibility of divisors in some tori and lattices

Let $E$ and $E'$ be two general elliptic curves. We consider the $2$-dimensional torus $A:=\frac{E\times E'}{(u\times u')\left((\mathbb{Z}/2\mathbb{Z})^2\right)}$, where ...

**15**

votes

**2**answers

273 views

### Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...

**2**

votes

**0**answers

97 views

### Reference request for the relation of Ext groups and bar construction

I need a reference for the description of Ext groups in mixed categories (i.e. abelian categories with a weight filtration and semisimple graded quotients) by using the bar complex, as mentioned in ...

**2**

votes

**0**answers

123 views

### A morphism of elliptic schemes that preserves the identity is a homomorphism

I am trying to understand the proof of the fact that any morphism $f \colon E_1 \rightarrow E_2$ of elliptic curves over an arbitrary base scheme $S$ satisfying $f(0) = 0$ must respect the group ...

**2**

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**0**answers

100 views

### Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...

**2**

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**0**answers

101 views

### Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...

**8**

votes

**1**answer

1k views

### why isn't the mobius band an algebraic line bundle?

When I hear the phrase "line bundle" the first thing that pops into my head is a mobius band. But this is a bad picture from an algebraic point of view since any line bundle on an affine variety is ...

**4**

votes

**0**answers

78 views

### Does a semistable curve descend to a regular base?

Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that
$f$ is proper, flat, and of finite presentation;
...

**2**

votes

**1**answer

147 views

### Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.
If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...

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votes

**0**answers

85 views

### When does a hyperelliptic Riemann surface admit a map of degree 3

Let $X$ be a hyperelliptic curve of genus $g>1$.
For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$?
I think a genus two curve $X$ admits a map of degree $3$.
Proof: Pick $P$ ...

**9**

votes

**1**answer

468 views

### Conditions for “bootstrapping” a smooth DM stack?

In the preprint "Smooth toric DM stacks", Fantechi, Mann and Nironi define the stacks of their title, and show that each of these can be obtained through the following sequence of steps:
1) start ...

**9**

votes

**0**answers

161 views

### Do canonical stacks exist over Spec(Z)?

Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism ...

**8**

votes

**0**answers

333 views

### What is the relationship between these two notions of “period”?

The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...

**5**

votes

**2**answers

1k views

### What is the Theorem of the Cube?

What is the "theorem of the cube" for abelian varieties? What is the statement and how should I think about it?

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**0**answers

78 views

### Coefficients of Hilbert polynomials

Recall that we can define the Hilbert series of a graded commutative algebra
$$\displaystyle S = \bigoplus_{n \geq 0} S_n$$
over a field $K$ by
$$\displaystyle \mathcal{H}_S(t) = \sum_{n=0}^\infty ...

**8**

votes

**1**answer

430 views

### Can one check formal smoothness using only one-variable Artin rings?

Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...

**8**

votes

**3**answers

649 views

### How to topologize X(R) when R is a topological ring?

Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...

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votes

**3**answers

601 views

### If $\Omega_{X/Y}$ is locally free of rank $\mathrm{dim}\left(X\right)-\mathrm{dim}\left(Y\right)$, is $X\rightarrow Y$ smooth?

Suppose I have a morphism $f:X\rightarrow Y$ such that the relative sheaf of differentials $\Omega_{X/Y}$ is locally free. Does it follow that $f$ is smooth?
The answer is no, but for a silly reason. ...

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votes

**0**answers

43 views

### Extending a model to a given compactification of its generic fiber

Let $R$ be a discrete valuation ring and $K$ its field of fraction. Let $X$ be a proper $K$-variety, $U$ a dense open and consider an $R$-model $\mathcal{U}$ of $U$.
Can we embed $\mathcal{U}$ in a ...

**3**

votes

**0**answers

81 views

### References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...

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votes

**1**answer

455 views

### Category of motivic spectra

When the survey Axiomatic Stable Homotopy, Neil Strickland, 2004 was written the category of motivic spectra was not investigated from the point of view of axiomatic stable homotopy, as considered ...

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votes

**0**answers

43 views

### Complementary polynomials

Denote $S=\{0,1\}^n$.
$\mathsf{MLP}_{d,n}=\{p\in\Bbb R[x_1,\dots,x_n]:p\mathsf{\mbox{ is mutilinear with total degree}}(p)=d\}$.
Is there an $n\geq d^2+1$ such that there exists distinct polynomials ...

**0**

votes

**1**answer

91 views

### Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...

**5**

votes

**0**answers

88 views

### Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...

**1**

vote

**1**answer

214 views

### linear section of codimension $k+1$ of a variety of dimension $k$

Let $X$ be a variety of dimension $k$ and degree $d$. If $L$ is a linear subspace of codimension $k+1$ such that $|L\cap X|$ consists of a finite number of points, is there a way to find the maximum ...

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**0**answers

71 views

### Moduli space of points - Gorenstein ideal

I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d).
I'm more interested in the algebraic point of ...

**60**

votes

**2**answers

4k views

### Is it known that the ring of periods is not a field?

I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...

**43**

votes

**2**answers

2k views

### What are reasons to believe that e is not a period?

I hope this rather soft question is suitable for MO, otherwise please migrate it to MSE.
In their paper defining periods [1], Kontsevich and Zagier without further comment state that $e$ is ...

**11**

votes

**1**answer

381 views

### Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.

**0**

votes

**0**answers

67 views

### What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?

I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex
$$
V ...

**0**

votes

**1**answer

148 views

### Stability conditions of coherent sheaves on abelian 3-folds

My work for now consists on understanding stability conditions of coherent sheaves on abelian 3-folds. I have the book by D. Huybrechts (the geometry of moduli spaces of sheaves), But I would like to ...

**-2**

votes

**0**answers

83 views

### Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris [on hold]

Let π:C′→C an unramified double cover of a complex Riemann surface C of genus g. With the symbol Nmπ we mean the norm application that takes a meromorphic function on C′ named h and produce a ...

**-2**

votes

**0**answers

67 views

### involution of a riemann surface [on hold]

I've have found in the book GEOMETRY OF ALGEBRAIC CURVES by Arbarello, Cornalba, Griffiths that if π:C′→C it's a double unramified cover of a complex riemann surface named C that we can define the ...

**1**

vote

**1**answer

427 views

### Minimal polynomial of symmetric endomorphism on abelian variety

Let $(A,\Theta)$ be a principally polarized abelian variety over an algebraically closed field $k$, and let $f$ be a symmetric endomorphism of $A$ (that is, $f^\dagger=f$ where $\dagger$ denotes the ...

**0**

votes

**0**answers

56 views

### Circle squarer and solution of polynomial equations [on hold]

Given a circle squarer, i.e. something that gives a length of $\pi$ given an unit length and vice versa, and a straightedge and compass, is it possible to solve any polynomial equation with rational ...

**1**

vote

**1**answer

284 views

### Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...

**4**

votes

**0**answers

90 views

### Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...

**3**

votes

**3**answers

724 views

### Poll about your proof of resolution of singularities and a request for advice

The questions first: What is the proof of resolution of singularities that you know?
Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of ...

**2**

votes

**0**answers

54 views

### Does the invariant from resolution of singularities provide a Whitney stratification?

The topic of Whitney stratifications came up in a lecture, and the general procedure in the examples was to decompose the singular locus of the variety into the strata starting with the "worst" ones. ...

**0**

votes

**0**answers

138 views

### What is a Beilinson spectral sequence?

I'm writing to ask just a question. I would like to understand better what is the Beilinson's spectral sequence and how it can be used. Is there any useful and clear reference you advice to someone ...

**18**

votes

**5**answers

1k views

### Is the category of commutative group schemes abelian?

I think, because in the category of schemes, all finite limits exist, the commutative group
objects with homomorphisms should form an abelian category.
Is this true? And do you know anywhere to cite ...

**10**

votes

**1**answer

187 views

### Is there a solvable point on any variety over the field of complex rational functions?

Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
Must $V$ have a solvable point?
The variety $V$ is assumed ...

**3**

votes

**1**answer

172 views

### Abelian varieties as analytic manifolds

Assume we have an Abelian varieties over the p-adic numbers, namely $
k=\mathbb{Q}_p$. Then the question is whether $A(k)$, the rational points over $k$, will form a p-adic analytic manifold.
I am ...

**32**

votes

**4**answers

3k views

### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

**2**

votes

**2**answers

190 views

### Rigid curves, and the “richness” of their homology class

Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H_2(X,\mathbb Z).$$
I have a very vague question about this situation:
Q. ...

**4**

votes

**1**answer

148 views

### Verlinde Formula and Theta Function Identities

The paper Fusion rules and modular transformations in 2D conformal field theory by Erik Verlinde mentions a simple case of rational conformal field theory, where the fusion algebra is just ...

**1**

vote

**1**answer

139 views

### rational point of a curve [on hold]

Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group ...