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

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### Bertini type theorem in positive characteristic

Let $f:X \to Y$ be a morphism of finite type over an algebraically closed field of characteristic $0$. Assume that $Y$ is irreducible and non-singular. Let $x \in X$ be a closed point and $T_xf:T_xX ...

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69 views

### A question about equivariant sheaves

Suppose we have an G-equivariant sheav $\mathcal F$ on a smooth variety $X$. Can we split $\mathcal F$ as sum of eigensheaves? (I have seen this for structure sheaf but not sure if we can do it for ...

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68 views

### Tautological line bundle after blow-up

Let $X$ be a projective manifold, and $Z$ be a submanifold of $X$ with codimension at least 2. Let $Y$ be the blow-up of $X$ along $Z$ with the exceptional divisor $E$. Question: what is the relation ...

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**1**answer

275 views

### Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays the sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classse as
$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; ...

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101 views

### A basic question on local cohomology

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow.
Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed ...

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**1**answer

206 views

### When does $R [x]/I $ have infinitely many idempotents?

Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotent and an unextended ideal $I $ in $R [x] $ such that $R[x]/I$ has ...

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775 views

### Grothendieck's “La longue Marche à travers la théorie de Galois”

It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.
Is there any way to obtain a copy (online or not) of "La ...

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44 views

### Does the action of a 2-torsion line bundle on $Pic^d(C)$ fix the number of sections?

Let $C$ be a smooth projective curve over $\mathbb{C}$. Let $A$ be a degree $d$ line bundle on $C$, and $M$ be a degree 0 line bundle on $C$ such that $M^2=\mathcal{O}_C$, that is, it is a 2-torsion ...

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52 views

### What is known about topological equivalence of polynomial dynamical systems on two different domains in R^n?

The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).
Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study ...

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151 views

### Quantum cohomology of line bundles over $\mathbb P^N$

Let $n,N$ be two positive integers. Consider the total space of the line
bundle $\mathcal O(-n)$ on $\mathbb C\mathbb P^N$. This is an algebraic variety with an action of $G=GL(N,\mathbb C)\times ...

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163 views

### Veronese embeddings of elliptic curves in weighted projective space

Let $E$ be an elliptic curve and $D_k=kp$ a divisor on $E$, where $p\in E$, for $k\in\mathbb{N}$.
Then we can reconstruct $E$ from the graded ring $R(D_k)=\bigoplus_{n\geqslant0}\mathcal{L}({nD_k})$: ...

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83 views

### Condition on moment polytope for a toric manifold to be Fano

Suppose $M$ is a symplectic toric manifold. This means there is a compact torus
$T$ that has a Hamiltonian action on $M$, with moment map $\mu:M \to \mathfrak t^*$, and $\dim(M)=2\dim(T)$. Can one ...

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79 views

### model theory of non-reduced schemes

In model theory one studies Boolean algebras of definable sets of complete theories. For many theories definable sets are in direct correspondence with geometric objects, for example, definable sets ...

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707 views

### What is the role of projective spaces in GAGA?

The GAGA theorem is a celebrated elaboration of the idea that complex analytic and complex algebraic geometry are equivalent, at least for smooth projective varieties/manifolds.
I am aware why this ...

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72 views

### A Nodal curve embedded in a smooth variety, is always regularly embedded?

I will consider more simple situation. Let C be a nodal curve which is a union of two $\mathbb{P}^1$, $C_1,C_2$. Which meets at a node $p$.
Consider $C$ is embedded in a smooth variety $Y$. Assume ...

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73 views

### how to prove that $k[XY]/(XY)$ is a $k[Z]-$ module finitely generated? [on hold]

I'm wonrking in this question: Let $A=k[X,Y]/(XY)$, and take the homomorphism $k[Z]\rightarrow A$ given by $Z\to X+Y$. Show that this homomorphism is finite and injective.
For the injectivity I can ...

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269 views

### Embedding of a proper scheme into a smooth one

Let $\mathbb{K}$ be any field and let $X$ be a proper $\mathbb{K}$-scheme. Does it exist a smooth, proper $\mathbb{K}$-scheme $Y$ and a closed immersion from $X$ to $Y$? This is tautological if $X$ is ...

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**1**answer

94 views

### Vector bundle with a perfect pairing and ($\mathbb Z/2$, $SL_r$)-bundle

I think this is a well knowing result but I can't find any reference,
Let $(E,q)$ be a vector bundle with a non degenerated quadratic form $q:E\rightarrow E^*$ with trivial determinant, suppose ...

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227 views

### Scheme of irreducible components

Let $\pi:X \to S$ be a morphism of schemes (I can assume that $\pi$ is sufficiently nice, e.g. proper and flat, but certainly not smooth).
Does there exist a scheme $I_{X/S}$ which parametrises ...

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284 views

### rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...

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79 views

### what's the minimal embedding of orthogonal grassmannian

Suppose X is the orthogonal grassmanian. We know the plucker embedding does not span the whole background CP^N, just span Subspace CP^m. My question is that is there an expression of the isometric ...

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120 views

### Points on algebraic varieties [on hold]

Suppose that $V$ is a geometrically integral algebraic variety, defined over some number field $K$. More concretely, we can think of $V \subset \mathbb{A}^n(K)$ for some positive integer $n$. Here ...

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112 views

### $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$?

Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental methods, that if $\nabla$ is an integrable connection on a vector bundle $L$ on $A$ ...

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106 views

### Affine variety wrought out of irreducible cubic polynomial in two variables, what does it look like?

Let $h \in \mathbb{R}[x, y]$ be an irreducible cubic polynomial. Consider the affine variety$$\{(x, y) \in \mathbb{R}^2: h(x, y) = 0\}.$$What are the qualitatively different possibilities for what ...

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156 views

### Exists $f \in I(X)$ such that $f(x) \neq 0$, $f(y) \neq 0$

Let $X \subseteq \mathbb{A}^n$ be algebraic, and let $x$, $y \in \mathbb{A}^n - X$. How do I see that there exists $f \in I(X)$ with $f(x) \neq 0$ and $f(y) \neq 0$.

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85 views

### Loci in the moduli space of K3 surfaces associated to lattices

The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a ...

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139 views

### Chow group over function field and algebraic equivalence

It is known that for smooth projective varieties $X,Y$ over $k=\bar k,$ $$CH^d(X_{k(Y)})=\varinjlim_{U\subset Y\ open}CH^d(X\times_k U)$$
I was wondering whether there was such an equality with ...

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77 views

### about transverse complete intersection

There are several questions about transverse complete intersection arising from L. Guth's paper:
http://www.ams.org/journals/jams/0000-000-00/S0894-0347-2015-00827-X/home.html
We say a polynomial ...

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110 views

### Can relative flatness of a sheaf be tested using (faithfully) flat morphisms?

Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O_Y$-modules $F$.
Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ ...

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58 views

### Rational connectedness of certain subvarieties of the linear series

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $|\mathcal{O}_X(a)|$ be the complete linear system for some integer $a>0$. Ofcourse, a general element of the linear system is a ...

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80 views

### Dual involution on the $Ext^1$

Let $X$ be a smooth algebraic curve over $\mathbb C$, and let $F$ be a vector bundle on it of degree $1$, take the dual of an extention $$0\rightarrow F^*\rightarrow E\rightarrow F\rightarrow0$$ is ...

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388 views

### Why would one “attempt” to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...

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138 views

### studying combinatorics [closed]

I am an undergrad, math major, and I had basic combinatorics class before (undergrad level.) Currently reading Stanley's Enumerative Combinatorics with other math folks. We have found this book ...

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145 views

### Hodge structure of the cohomology of a complement

What is the hodge structure given to the cohomology of the complement of a closed subset with respect to a smooth variety?
This is not quite what is wanted. In fact,
The hodge structure is ...

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80 views

### Direct limit of coherent sheaves and semi-stability

Let $R$ be a discrete valuation ring, $\{B_i\}_{i\in I}$ be an inductive system of $R$-algebras of finite type and $B$ the direct limit of the inductive system. Let $X$ be a regular, projective ...

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44 views

### cell decomposition of grassmannian [closed]

What is the cell decomposition of grassmanian?suppose it's p dimensional subspaces in p+q dimensional vector space. I know there is a piece C^pq, what is the codimension 1 sub varieties? How many ...

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150 views

### Reducibility of a variety

Suppose $X$ is a variety defined in $\mathbb P^n\times\mathbb P^n$ by a divisor. It projects down to first $\mathbb P^n$ almost every fiber is not $\mathbb P^n$ and assume every fiber is reducible. ...

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156 views

### Geometric meaning of conductor

Supppose $L/K$ is a finite extension, choose $\theta \in O_L$ such that $L=K(\theta)$. We define the conductor of ring $O_K[\theta]$ to be an ideal of $O_L$, namely: $F=\{\alpha\in O_L|\alpha\cdot ...

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300 views

### vanishing theorem in algebraic geometry

This is a general question: As we know there are a lot of vanishing theorems like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those ...

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116 views

### Constancy on large subsets

Let $X$ be a smooth proper variety of dimension greater than one defined over a number field. Let $A_i$ be a sequence of pairwise disjoint finite closed subsets of $X$ indexed by natural numbers ...

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89 views

### the fundamental theorem in algebra [closed]

it is well known that any polynomial with complex coefficients and degree n has exactly n complex roots . The question is is this case can be true when we move to the p-adic domain?
Thank you in ...

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172 views

### Mapping class group action on fundamental group of punctured elliptic curves

Let $(\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}}$ be the moduli stack of elliptic curves over $\overline{\mathbb{Q}}$. By Oda, we know that its etale fundamental group is $\widehat{SL_2(\mathbb{Z})}$.
...

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### algebraic numbers in the p-adic domain [closed]

I would like to know the relation between the conjugates of sum algebraic numbers and the sum of conjugates of these algebraic numbers. Thank you.
Amran dalloul, Phd student

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108 views

### For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?

Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...

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280 views

### Is the Brauer group functor a Zariski sheaf?

Is the functor $$\operatorname{Br} : \operatorname{Sch}^{\operatorname{op}} \to \operatorname{Ab}$$ sending $X \mapsto \operatorname{Br}X$ a sheaf for the Zariski topology on $\operatorname{Sch}$?
...

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134 views

### When is the base change map for Ext an isomorphism?

In the first section of Altman and Kleiman's paper "Compactifying the Picard scheme", a base change map for Ext sheaves is defined. I am interested in knowing when this map is an isomorphism.
I ...

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145 views

### Brauer group of a rational variety

This is a follow-up question to this question. There and here $X$ is a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. My question is:
...

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77 views

### When is an intersection of quadrics empty

Is there an easy way to tell when the intersection of a set of quadrics (in complex projective space) is empty? Assume you know the quadrics explicitly.

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120 views

### Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution?

I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}_n$. ...

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212 views

### $(\mathbb{CP}^1)^n/S_n \overset{\sim}{\to} \mathbb{CP}^n$ [closed]

Can someone refer to me a source that describes the construction of a homeomorphism $$(\mathbb{CP}^1)^n/S_n \overset{\sim}{\to} \mathbb{CP}^n?$$I am not an algebraic geometrer and I would like to see ...