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

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**1**answer

205 views

### Continued fraction expansion of an algebraic number and its conjugates

Let $w$ be an element of a Galois extension $L:\mathbb{Q}$ such that $\text{Gal}(L/\mathbb{Q})=\langle g\rangle$ is cyclic of order $n$ (here $\mathbb{Q}$ is rationals). Suppose we know the continued ...

**1**

vote

**1**answer

93 views

### Boundedness of the number of curves negative on a varying big divisor

For a divisor $D$ on a smooth complex projective surface $X$, the stable fixed part is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the ...

**5**

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**0**answers

123 views

### Is every Noetherian *connected* ring a quotient of a Noetherian domain?

This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here.
Geometrically, this asks if every Noetherian connected affine scheme can be ...

**4**

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**0**answers

102 views

### Global geometry of discriminant locus

Let $X$ be a smooth projective threefold, and $\pi : X \to S$ an elliptic fibration over a surface (i.e. flat, with general fiber an elliptic curve). I'm interested in constructing such fibrations ...

**0**

votes

**1**answer

80 views

### A semi-ampleness criterion for homogeneous bundles on homogeneous spaces?

Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a ...

**2**

votes

**1**answer

111 views

### Rational normal curves on Grassmanians

Consider the Grassmanian $G(k,n)$ ($k\le \frac{n}{2}$) and take its Plucker embedding. Consider now the space of all normal rational curves of degree $k$, contained in the Plucker embedding of the ...

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vote

**1**answer

104 views

### Neron model: can number of components decrease after based change?

Suppose I have Neron model over some discrete valuation ring.
Is there a result such that the number of components of the fiber over the closed point cannot decrease after some based change?
In ...

**1**

vote

**1**answer

97 views

### Generic deformation of Hilbert scheme of points on K3 surface

i was thinking about deformations of hyperkahler manifolds, in particular hilbert schemes of points on K3 surfaces and I think I realized something. I'm here to ask you if I'm right.
Take $X^{[n]}$ ...

**20**

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**0**answers

447 views

### History of the Proj construction in algebraic geometry

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...

**3**

votes

**1**answer

137 views

### Flat family with special fiber $\mathbb{C}\mathbb{P}^1$

Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to ...

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**0**answers

479 views

### Has there been any progress on the standard conjectures on algebraic cycles?

What's the current state of these conjectures?
Who is working on them?
In "Standard conjectures on algebraic cycles" Grothendieck says:
"They would form the basis of the so-called "theory of ...

**9**

votes

**2**answers

254 views

### Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?
To prevent things from being too easy, I ...

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**1**answer

128 views

### Pull-Back of an ample line bundle

here is my question:
I work over the field of complex numbers, and my schemes are separated and of finite type. Let $X$ be a quasi projective scheme (but not projective), let $Y$ be a regular ...

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votes

**1**answer

143 views

### Holomorphic Foliations having transverse sections

In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation ...

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vote

**1**answer

280 views

### When flatness of a morphism implies smoothness?

EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth.
Is it true that there exists a ...

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vote

**1**answer

63 views

### Semicontinuity of degree of fibers for a proper map

Let $f:X \rightarrow T$ be a proper morphism from a projective scheme $X$ to a smooth projective curve $T$ over $\mathbb{C}$. I know that fiber dimension is upper-semicontinuous, but is the degree of ...

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**0**answers

147 views

### Blow-up and tangent cone

Let
$$ f\colon X \rightarrow Y$$
be a (possibly non-dominant) morphism of complex varieties which is birational with its image. Assume $f$ contracts a sub-variety $E$ of $X$ to a point $p$ of $Y$; ...

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**0**answers

180 views

### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...

**1**

vote

**1**answer

158 views

### Classification (and automorphisms) of torsion-free modules/sheaves

I would like to know what can be said about the classification of torsion-free modules.
For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does ...

**5**

votes

**0**answers

128 views

### Strengthening of Suslin's rigidity argument?

To fix the situation, let $k$ be an algebraically closed field, and let $C$ be a smooth projective curve over $k$. Suslin's rigidity argument implies in particular that any class in ...

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**1**answer

295 views

### Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that ...

**5**

votes

**1**answer

104 views

### Transfers on Bloch groups and scissors congruence groups

I have a couple of questions concerning existence and description of
transfers for Bloch groups and scissors congruence groups/pre-Bloch
groups.
To fix notation and recall definitions:
From the ...

**0**

votes

**1**answer

155 views

### is grassmannian rational connected or not [closed]

I wan to know if Grassmannians are rational connected? Any reference describe how to tell if a variety is rational connected or not?

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**1**answer

343 views

### Maps to projective space == line bundles; what do maps to weighted projective space correspond to?

A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a weighted projective ...

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**0**answers

72 views

### singularities preserved by integral closure

Let $X$ be an affine variety. Let $A$ be the coordinate ring of $X$ and let $K$ be the fraction field of $A$. Given a Galois extension $K\subset L$, let $B$ be the integral closure of $A$ in $L$. Let ...

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**1**answer

126 views

### Can someone tell me properties of Douady space?

I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of ...

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votes

**1**answer

201 views

### Articles about Weil cohomology theory by Grothendieck and Artin

In "The Standard Conjectures" Kleiman says that the following properties of Weil cohomology theory were proved in 1963 for étale cohomology by Artin and Grothendieck, except for the last one that it ...

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**1**answer

263 views

### Is there any publication of “Beilinson’s dream” on motivic (complexes of) sheaves?

In "Standard conjectures of algebraic cycles" nLab says:
"... They were also followed by “Beilinson’s dream” on motivic (complexes of) sheaves which comprise so called standard conjectures of ...

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votes

**1**answer

92 views

### Sabbah b-functions factoring

Sabbah defined a version of b-functions for multiple functions in his 1987 paper here: http://archive.numdam.org/ARCHIVE/CM/CM_1987__62_3/CM_1987__62_3_283_0/CM_1987__62_3_283_0.pdf
According to ...

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**1**answer

94 views

### Cohomology of a fibered surface

Let $R$ be a complete Henselian discrete valuation ring, $\pi:X \to \mathrm{Spec} (R)$ be a smooth, proper, integral, flat $\mathrm{Spec} (R)$-scheme of dimension $2$. Assume that the genus of the ...

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**1**answer

174 views

### Moduli of curves in characteristic zero

Let $K$ be a field of characteristic zero, and let $\overline{K}$ be its algebraic closure. Let $\overline{M}_{g,n}(K)$ and $\overline{M}_{g,n}(\overline{K})$ be the coarse moduli spaces parametrizing ...

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**0**answers

211 views

### Does the following object has a name in algebraic geometry?

Suppose $X$ is a projective variety and $D$ is a smooth divisor and let $L = \mathcal{O}(D)$ be the line bundle corresponding to $D$. Consider $X \times \mathbb{P}^1$ with the line bundle ...

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**2**answers

209 views

### Lifting to char 0, references and questions

Suppose that I have a surface $S$, smooth proper over an algebraically closed (perfect?)field $k$ that lifts algebraically to some $S_W$ defined over a field of char 0. I am interested in properties ...

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**0**answers

67 views

### A question related to the Grauert semi-continuity theorem

Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on ...

**2**

votes

**1**answer

118 views

### Fibers of the Bott-Samelson Resolution of Schubert Varieties

Is there an explicit (perhaps visual) description of the fibers of the Bott-Samelson Resolutions of Schubert Varieties? Let's fix $G$ to be $GL_n(\mathbb{C})$.
Also, how would the answer to the ...

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**0**answers

95 views

### Symmetric function of residues

Let $X$ be a smooth projective irreducible curve over an algebraically closed field $k$, and let $\omega$ be a rational 1-form on $X$, so $\sum_x {\rm{res}}_x(\omega) = 0$ (sum over $x \in X(k)$). For ...

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**0**answers

96 views

### Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) ...

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**0**answers

601 views

### New(?) reciprocity law

Consider three functions $f, g$ and $h$ on a smooth curve $X$ over $\mathbb{C}$.
I have found the following equality:
$$\sum (res(f\frac{dg}{g})\frac{dh}{h}-res(f\frac{dh}{h})\frac{dg}{g})=0.$$
Here ...

**1**

vote

**1**answer

144 views

### how do you define the sheaf of Teichmuller level structures when a section does not exist?

In section 5.5 of Deligne + Mumford's paper "Irreducibility of the Moduli Space of Curves", they introduce the notion of Teichmuller level structures.
You can find the paper here: ...

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votes

**2**answers

146 views

### First cohomological support locus of a fibration

I have a fibration $f \colon S \longrightarrow E$, where $S$ is a compact, complex surface of general type belonging to a special class I'm studying and $E$ is an elliptic curve.
I computed the ...

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**0**answers

35 views

### quotient singularities and direct images of resolution map

We are working in the complex case, so over $\mathbb{C}$ :
$W\rightarrow W_{1} \rightarrow Z \rightarrow Y$,
map $g$ between $Y$ and $Z$ is a semistable family (normal crosings and reduced fibers) ...

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votes

**1**answer

123 views

### Minimum length path touching $n$ circles

Given $n$ non-overlapping circles of radius $1$ (i.e., the distance between the centers of any two circles is greater than $2$), how to find the minimum length path (the path can be of any form) that ...

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**1**answer

204 views

### Generalizations of de Franchis and function field Mordell

The classical de Franchis theorem, as generalized by S. Kobayashi and T. Ochiai ("Meromorphic mappings onto compact complex spaces of general type," Inventiones, 1975), states that if $X$ is a complex ...

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**1**answer

191 views

### smooth quotient out of a singular variety?

If $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $G$ is a finite group acting faithfully on $X$, then the Shepard-Todd theorem gives us some criterion for $X/G$ to be smooth.
My ...

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votes

**1**answer

159 views

### Regular singularities and the infinitesimal site

Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$.
A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent ...

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**0**answers

130 views

### Coherent sheaves and Mitchell's embedding theorem

Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...

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**0**answers

81 views

### Quotient singularities and higher direct images

$W\rightarrow W_{1} \rightarrow Z \rightarrow Y$,
map $g$ between $Y$ and $Z$ is a semistable family (normal crosings and reduced fibers) of n-folds over a smooth curve Y, then the
map between ...

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**0**answers

489 views

### Degrees of maps from curves to $\mathbb P^1$

Let $a$ and $b$ be two relatively prime natural numbers. What is the largest number $c$ such that there is a curve with maps to $\mathbb P^1$ of degree $a$ and $b$ but no map to $\mathbb P^1$ of ...

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votes

**2**answers

170 views

### Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...

**0**

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**1**answer

94 views

### rank of Abelian schemes under ample hypersurface section

Let $k$ be an algebraic closure of a finite field, $\ell \neq \mathrm{Char}(k)$ be prime, $S/k$ a smooth projective geometrically connected surface and $C/k$ a smooth ample connected hypersurface ...