# Tagged Questions

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

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88 views

### Quadrics cutting out a polygon

Let $l_1,l_2,l_3,l_4,l_5\subset \mathbb P^4_k$ be distinct lines such that $|l_i\cap l_{i+1}|=1$ for all $i\ mod\ 5$ and $l_i\cap l_j\neq \emptyset\iff j=i+1\ mod\ 5$ (so that $\cup_{i=1}^5l_i$ is a ...

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105 views

### self intersection of a curve in a surface

Suppose $S$ is a compact complex surface, $C\subset S$ is a one dimensional irreducible subvariety (a curve). Suppose further, there exists a family of biholomorphism of $S$ nearby the identity map. ...

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**0**answers

108 views

### Polynomials with some roots whose product is 1

I asked this question in this post but have not got a full answer. So I post it again on MO.
Consider the complex coefficient polynomial equation
\begin{eqnarray}
...

**4**

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**0**answers

89 views

### Deligne-Hitchin twistor space

Let $M$ be the moduli space of stable holomorphic Higgs bundles on a Riemann surface $\Sigma$. By Hitchin s work it is equipped with a hyper-kaehler structure, and one can define its twistor space, a ...

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149 views

### Flat connection from gauged WZW model

$\newcommand{\g}{\mathfrak g}$
$\newcommand{\h}{\mathfrak h}$
In short my question is :
Has someone worked out the flat connection that one should get from the gauged WZW model in genus 0 ?
Some ...

**7**

votes

**1**answer

203 views

### Fiberwise criterion for a stack to be a gerbe

Let $f:X\to Y$ be a morphism of algebraic stacks.
If the geometric fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces.
I'm wondering about analogues of this fiberwise ...

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61 views

### linearly equivalence of line bundles

Suppose we have a complex manifold $X$ with a fmaily of nontrivial holomorphic biholomorphisms $F(t,x)$ where $t\in\mathbb C,|t|<1$. And $F(0,x)=Id$ and $F(t,x)$is holomorphic in all variables. ...

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85 views

### Image of the smooth locus under a GIT quotient

Let $M$ be an irreducible component of a GIT quotient $Q//PGL(m)$, where $Q$ is a reduced quasi-projective scheme over an algebraically closed field. Is it possible for the image of the smooth locus ...

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**2**answers

222 views

### Example of a non-Kähler manifold with varying plurigenera

Let $X \stackrel{\pi}{\to} \mathbb{D}$ be a proper holomorphic family with fibres $X_t = \pi^{-1}(t)$. Siu proved, when the $X_t$'s are projective, that the plurigenera $h^0(X_t, mK_{X_t})$ are ...

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98 views

### Finite groups as intersection of algebraical groups [migrated]

Well known that any finite number of points can be seen as intersection of two algebraical curves. Is it true that any finite group $G$ can be seen as intersection of two (connected) one dimensional ...

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votes

**3**answers

921 views

### Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...

**30**

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**1**answer

642 views

### A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
In an effort to ...

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**1**answer

206 views

### Question related to Fermat curve: Does the equation $A x^n + By^n = C z^n$ have any solution in $\mathbb{N}$?

Let $A, B, C \in \mathbb{N}$ be such that $\gcd(A,B,C)=1$. Is it known if the equation $A x^n + By^n = C z^n$ has any non-trivial solutions $x,y,z \in \mathbb{N}$? I know there are no such solutions ...

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**0**answers

184 views

### What is Koszul dual of a curve?

Let $X$ be a curve embedded into a projective space $\mathbb P$ such that
it is cut out (scheme-theoretically or ideal-theoretically) by quadrics.
What is known about the Koszul dual of the ...

**-1**

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44 views

### Order of Contact for a general tangent line of a cubic threefold [migrated]

I am trying to solve Exercise 18.21 of Harris "Algebraic Geometry".
In the proof of the unirationality of a smooth cubic threefold X he claims that a general tangent line to X at a general point p ...

**4**

votes

**1**answer

145 views

### infinitesimal lifting criterion for non-noetherian schemes

We have the "standard criterion" which says that a morphism $f:X\rightarrow Y$ is smooth if:
1/ $Y$ is locally noetherian.
2/ $f$ is locally of finite type and satisfies lifting criterion for ...

**3**

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**1**answer

154 views

### Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$

Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks.
My guess is that $G$-gerbes for $G$ an abelian ...

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**0**answers

70 views

### How order of divisor with support at infinity is changed at reduction?

Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following
The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$.
To ...

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90 views

### Chern classes of a family and Chern classes of a member

Let $X$ be a smooth projective variety over an algebraically closed field $k$ and $\mathcal E$ a family on torsion free coherent sheaves on $X$ parametrized by a smooth curve (over $k$) i.e. a ...

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votes

**3**answers

753 views

### Does anyone know the classification of fourth order surfaces?

Does anyone know the classification of fourth order surfaces? By "fourth order surface" I mean a surface defined by an equation of the form $$f(x, \, y, \, z)=0,$$
where $f$ is a polynomial of degree ...

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**0**answers

134 views

### Dynamical Mordell-Lang on Kahler manifolds?

Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...

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**1**answer

192 views

### Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?

Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$.
We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ ...

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**1**answer

130 views

### The principal bundle of embeddings

In a paper of P. Michor, it was shown that Emb(M,N) is a smooth principal diff(M)-bundle, M and N are smooth locally compact manifolds provided dim M < dim N. My question is why there is a ...

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**1**answer

342 views

### higher algebraic homotopy groups for schemes?

I think I understand how to define the algebraic fundamental group $\pi^{alg}_{1}(X)$ of a scheme and I think I understand the relation between $\pi^{alg}_{1}(X)$ and $\pi_{1}(X(\mathbb{C}))$, where ...

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**3**answers

319 views

### When can the “homotopy exact sequence” of etale fundamental groups for a smooth curve fail to be exact?

Suppose you have a smooth proper curve $f : \overline{X}\rightarrow S$ over an arbitrary locally noetherian scheme $S$ with a section $e : S\rightarrow \overline{X}$. Let $X := \overline{X} - e(S)$. ...

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110 views

### Obstruction to lifting of global sections of invertible sheaves

Are there known examples of smooth projective hypersurface in $\mathbb{P}^3$, say $X$ and an invertible sheaf $L$ on $X$ with $H^0(X,L)>0$ satisfying the following property: There exists an ...

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**1**answer

238 views

### Mapping scheme from a proper variety

Let $X$ be a proper scheme over a field $k$. Let $T$ be a scheme over $k$. Is it true that morphisms $T \times X \to \mathbb{A}^1$ are in bijection with morphisms $T \to \Gamma (X, \mathcal{O}_X)$ ...

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207 views

### Can one prove the poincare duality for projective scheme by proving it for projective space?

It's well known the relationship between Poincare duality and Thom isomorphism（I mean cohomology purity $R^q i^! F=0$ if $q\neq c $ ) $\quad $
$Rf_!Ri_!=R(f|_Z)_!$ where f is $P_k^n\rightarrow k$ ...

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112 views

### Stable vector bundles in Weil's parametrization

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes ...

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**1**answer

230 views

### Analogue of Tate curve for $g>1$

Is there any analogue of the Tate curve for (principally polarized) abelian varieties of dimension $g$ ?

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**1**answer

106 views

### Smoothness and quotient

Suppose we have a smooth Mumford's quotient $Q//PGL_k(m)$ where $Q$ is a quasi-projective variety and $k$ is an algebraically closed field of positive characteristic. Is it true that $Q$ is also ...

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votes

**1**answer

399 views

### What are local spaces and what are they good for?

Factorization structures have been popular in the past decade. Recently a variant of this structure has been suggested by Ivan Mirkovic (and possibly collaborators). This variant, which goes under the ...

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**1**answer

74 views

### Pencil of singular affine hypersurfaces differing only in the constant term

Question--quick version: Does there exist a (nonconstant) polynomial $f \in \mathbb C[x,y,z]$ such that for all $c \in \mathbb C$, the affine hypersurface cut out by $f + c$ is singular?
Motivated ...

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**1**answer

195 views

### Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states ...

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**1**answer

257 views

### Intuition for Picard-Lefschetz formula

I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I").
To summarize the setup, we ...

**3**

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**1**answer

162 views

### Is there a covering of Prym variety?

$\mathstrut$Hi, guys!
Let $C$, $C^\prime$ be projective smooth irreducible algebraic curves over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$), $\phi : C$ $\to$ $C^\prime$ a ...

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votes

**1**answer

117 views

### When stable curves can be embedded to their Jacobian?

Let $X$ be a stable curve over a complete DVR $R$ with smooth generic fiber. If $X$ has a $R$-rational point, by the universal properties of Neron models, we obtain a morphism from $f: X-X_{s}^{\rm ...

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**2**answers

258 views

### Lifting the Hasse invariant mod $2$

Katz defines in Section 2.0 $p$-adic properties of modular schemes and modular forms the Hasse invariant as a mod $p$ modular form $A$ of weight $p-1$. In other words, it is a section of ...

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vote

**1**answer

91 views

### the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
If $n$ is even, then $det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...

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votes

**1**answer

107 views

### generic irreduciblity

Suppose we have a proper morphism $f:X\rightarrow Y$ and $0\in Y$. If the fiber $f^{-1}(0)$ is irreducible and reduced, is the set $\{y\in Y|f^{-1}(y) \text{ is irreducible and reduced}\}$ open?

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137 views

### Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question.
I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will ...

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**0**answers

128 views

### del Pezzo surfaces and exceptional algebras

It is well-known that $H^2(dP_n, Z)$ for a del Pezzo surface of degree $9-n$ which is $\mathbb{P}^2$ blown up at $n$ generic points, is encoded by the exceptional Lie algebra $E_n$. However, the Mori ...

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228 views

### Sporadic and Exceptional

I have been reading this recent paper of J.McKay and YH. He (they've written a number of papers recently, including a fun and joking one on 42 which overflow commented on) called "Sporadic and ...

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127 views

### ideals linked to an almost complete intersection

Is a grade $3$ type $3$ perfect ideal in a Noetherian ring linked to a grade $3$ almost complete intersection? I know that grade $3$ type $2$ perfect ideals are (by a work of Anne Brown).

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158 views

### Detecting $k$-affinoid spaces by vanishing cohomology

The property of being an affine scheme can be tested against all quasi-coherent sheaves in the following sense: a noetherian scheme $X$ is affine iff $H^i(X,\mathcal{F}) = 0$ for all quasi-coherent ...

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votes

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315 views

### History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar.
However, in this note by Lawvere the author writes:
"I myself had learned the ...

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**0**answers

154 views

### Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference
in the literature...
Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...

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votes

**1**answer

103 views

### Pullback of $0$-cycle by generically finite rational map

Let $f:Y\dashrightarrow X$ be a generically finite (and separable) rational map of smooth projective $k$-varieties and $x,y\in U(k)$ be two rational points, where $U\subset X$ is an open subset of $X$ ...

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vote

**1**answer

103 views

### Looking for a good exposition - Rees Construction

Can someone point me in the direction of a good exposition of the Rees Construction in Hodge Theory?
Thanks!

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**0**answers

98 views

### Finiteness of torsion in $\mathcal{K}_2$-cohomology

Let $F$ be a number field, $C$ be a smooth projective curve over $F$ and $\mathfrak{C}$ be a proper regular model. I am interested in $\mathcal{K}_2$-cohomology, i.e., Zariski cohomology of the sheaf ...