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

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55 views

### étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$.
Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$.
For ...

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32 views

### Positivity of holomorphic bisectional curvature and existence of rational curves

Let $X$ be a compact Kähler manifold such that $K_X$ is nef. When
there is no rational curve on $X$?
Motivation: Siu-Yau by the positive bisectional curvature assumption, produced a rational ...

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111 views

### Is there an explicit construction of Tate-Beilinson residue?

Background: Tate's construction of abstract residues is generalized in Beilinson's constructon as follows:
Let $E$ be a unital, cubically decomposed $k$-algebra, i.e., (a) there are two sided ideals ...

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**1**answer

124 views

### Regular minimal model of $X_0(p^2)$

Consider the compactified modular curve $X_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following:
Where do the cusps of $X_0(p^2)_{\mathbb{Q}}$ live? That ...

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167 views

### the origin of the differential of Spectral Sequence? [on hold]

I'm wondering why the differential in homological Spectral Sequences $(E^*_{p,q},d^r)$ is defined as;
$d^r_{p,q}: E^{r}_{p,q} \rightarrow E^{r}_{p-r,q+r-1}$ .
From Weibel's homological algebra(p122), ...

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92 views

### Kunneth decomposition of the relative diagonal of a projective bundle

Let $\mathcal{E}$ be a projective bundle of rank $r$ over a smooth complex quasi projective variety $B$, and form its associated projective bundle $\chi :=\mathbb{P}(\mathcal{E})$. Let $\pi : \chi ...

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101 views

### Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...

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**1**answer

161 views

### Bounding the degree of an algebraic extension containing solutions to polynomials

Also posted on math.stackexchange...
Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by ...

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**1**answer

111 views

### Density of non-algebraic leaves in the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has ...

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121 views

### Are these moduli problems of curves “well-behaved”?

Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.
Let $H_{X,d}$ be the Hilbert scheme ...

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**1**answer

163 views

### harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?

Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where
What happens in the $p$-adic case? Is there sphere ...

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**1**answer

135 views

### projectivity with assumption of big and semi-amplness

Let $X$ be a compact Kaehler manifold with $D$ be an effective divisor on $X$ such that $K_X+D$ is semi-ample and big then $X$ is projective?

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124 views

### $\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes

Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes?
Of course, smooth schemes that are $\Bbb A^1$-fibrant are ...

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**1**answer

312 views

### Normalization of complete intersection

Let $A$ be an integral complete local ring over a field which is complete intersection.
Let $B$ be a normalization of $A$.
Q. Is $B$ Gorenstein?
I guess that even the normalization of Gorenstein ...

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**1**answer

112 views

### Is the Grassmannian contained in a Plucker hyperplane?

Is the dimension of the projective space $\mathbb{P}^{{n \choose k} -1}$ into which we embed the Grassmannian $G(k,n)$ of $k$-planes in $n$-space minimal? In other words, is the Grassmannian variety ...

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**1**answer

117 views

### No cuspidal character sheaves on GL(n)

We need a reference for the fact that there are no cuspidal character sheaves on $GL_n$ unless $n=1$.
See page 11 of http://www.kurims.kyoto-u.ac.jp/~arakawa/Henderson_mgsctalk2.pdf.

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72 views

### Moebius linear fractional transformation in higher dimensions [closed]

A linear fractional transformation is a function of the form $f(z)=\frac{az+b}{cz+d}$ where $a,b,c,d \in\mathbb{C}$. If $ad-bc\neq0$ then $f$ is called Möbius transformation. The question is if there ...

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**1**answer

133 views

### Invariance of the fiber-dimension of a finite map

Let $A\subseteq B$ be commutative Noetherian rings such that $A$ is a regular ring, i.e., $A_{\mathfrak{m}}$ is a regular local ring for all maximal ideals $\mathfrak{m}$ of $A$ and $B$ is a finite ...

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**1**answer

225 views

### What are Motivic homotopy types?

There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it.
I would like to know the reference in which Grothendieck did it, ...

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**4**answers

812 views

### Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...

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55 views

### Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...

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**1**answer

163 views

### Extending a prime divisor to a principal divisor

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist ...

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**1**answer

124 views

### Minimal Nagata-like compactification

Working over a field $k$, Nagata's compactification theorem implies that any separated scheme $X$ of finite type over $k$ admits a compactification (a dense open immersion $i \colon ...

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114 views

### Abstract VFC vs. what people actually use for Quintic 3-fold

Moduli space of genus $0$ degree $d$ maps in a quintic Calabi-Yau threefold $X$, written as $\overline{\mathcal{M}}_{0,0}(X,[d])$, can be embedded in the corresponding moduli space of $\mathbb{P}^4$, ...

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**1**answer

369 views

### Holomorphic vector bundles over $\mathbb{C}^{n}\setminus 0$

Is it true that every holomorphic vector bundle over $\mathbb{C}^{n}\setminus 0$ is trivial? If not true, how can one construct a counterexample?
And just a small note here (wrong):
For $n\leq 2$, ...

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**1**answer

232 views

### Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?

Main Question: What Is the correpondence between flows and vector
fields in algebraic geometry?
Here is a more precise statement could be an answer If it was true (I have no idea it is):
...

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**1**answer

307 views

### Noetherian Rings in Constructive Mathematics

These definitions are largely from pages 92-93 of Ingo's notes. All rings are commutative with 1. I'm interested in understanding the extent to which discussion of Noetherian rings can be carried over ...

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248 views

### Cohomology of tangent sheaf of a singular hypersurface

Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p_i\in X$. We may assume that $X$ has ordinary singularities at the $p_i$'s.
Does there exists a formula, perhaps in ...

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**1**answer

232 views

### How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...

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**1**answer

409 views

### Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$.
In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...

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**1**answer

159 views

### The compatibility of the Gysin sequence with mixed Hodge structures

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$.
Then it is well known that the ...

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**1**answer

119 views

### A question on surjectivity of a bilinear quadratic map

Let $a=(a_0, a_1, ..., a_n )$, $b=(b_0, b_1, ..., b_n )$ that belong to ${\mathbb R}^{n+1}$. Define polynomials $f_a (t)=a_0 +a_1 t+ ... + a_n t^n$ and $f_b (t)=b_0 +b_1 t+ ... + b_n t^n$ and let ...

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427 views

### Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...

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70 views

### Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, ...

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**1**answer

708 views

### What is a field [Körper] really?

The notion of a field (a commutative ring $R$ with $0\neq 1$ and $R^\times=R-\{0\}$) seems to fit uncomfortably into modern algebra. To see what I mean, consider the following statements:
The ...

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**1**answer

162 views

### Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page.
I am asking if it ...

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**1**answer

192 views

### Algebraic points of uniformly bounded degree on an algebraic variety

Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number ...

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**1**answer

171 views

### Deformation of $\mathbb{P}^1 \times \mathbb{P}^1$

I learnt that $\mathbb{P}^1 \times \mathbb{P}^1$ is rigid, but can be deformed to a non-rigid Hirzebruch surface $S$. Suppose $\pi: M \to B$ is such deformation such that $\mathbb{P}^1 \times ...

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156 views

### K theory and derived categories

Some months ago I studied Beilinson's paper about generators for the derived category of $\mathbb{P}^n$, "Coherent Sheaves on $\mathbb{P}^n$ and problems of linear algebra".
As next step, I moved to ...

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**1**answer

410 views

### Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...

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70 views

### Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say
...

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103 views

### Finding the Chern Class of a the pushfoward of a invertible sheaf

I am trying to understand what happens to the Chern Classes of an invertible sheaf $F$ over a complete intersection reduced curve of genus $g$ and degree $d$, when viewed as a invertible sheaf of ...

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156 views

### Normal bundle to fibers of a rational morphism

Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...

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128 views

### Stronger version of Bertini's theorem

In char 0, is there a generalised version of Bertini's theorem that will ensure that for a proper map $f: Y\rightarrow X$ between smooth projective varieties and for every point $x\in X$ we can find ...

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190 views

### Characters of simply connected semsimple algebraic groups over local fields

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.
However, it is quite possible ...

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115 views

### Uniform bound on the Mordell-Weil rank of elliptic curves

I want to know that if there is an uniform upper bound for the rank of elliptic curves over $\mathbb{C}(t)$, the rational function field over complex numbers, and generaly over the function field of ...

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220 views

### j-invariants for isogenous elliptic curves

Let $E$ be a smooth complex elliptic curve, and $\sigma$ translation of $E$ given by a point $p$ on $E$ of finite order, with respect to some fixed origin.
What are the $j$-invariants related with ...

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**1**answer

142 views

### Methods of showing a variety is stably rational

As anyone who follows the algebraic geometry tag on arXiv will probably know, there has been a lot of papers recently showing various varieties are non-stably rational. What I am interested in however ...

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**2**answers

253 views

### Group cohomology of fundamental group of a curve

The question should be an elementary result in the theory of etale cohomology, but I failed to understand it because I am a complete beginner of the theory. So, I should apologise in advance for this ...

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207 views

### Construction of Dualizing sheaf

I was going through the construction of dualizing sheaf given in Hartshorne [III, 7, Lemma 7.4]. The proof apparently omits lots of details. In particular it does not mention any of the $i^* , i_*, ...