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

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4
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279 views

Use of derivators to the theory of motives?

This is a rather imprecise question but i think this could become a interesting pool of ideas and comments. The theory of motives has evolved to a complex field of research the moment Voevodsky (and ...
0
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1answer
160 views

An integrality theorem for immersions of complex projective spaces in the euclidean space

There are three questions: Please let me know your proof of the following theorem: If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to ...
0
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0answers
76 views

Crepant partial resolution of rational singularities

Let $X$ be a projective variety with at most klt singularities. If $K_X$ is Cartier, then the singularities are canonical (See answers in The minimal model program and symplectic resolutions for more ...
1
vote
1answer
129 views

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that ...
0
votes
1answer
191 views

The number of solutions of a Diophantine equation [closed]

Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity. That is, I am asking whether the number ...
10
votes
2answers
448 views

On a proposition in Hartshorne's paper “Ample vector bundles on curves”

In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following: Let $A$ be an abelian variety [over an alg. closed field ...
1
vote
0answers
68 views

About the reduceness of the commuting scheme associated with a symmetric pair

my question is the following one: Let $G$ be a connected reductive algebraic group over the field of complex numbers, and let $V$ be a linear representation of $G$ obtained as the isotropy ...
11
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0answers
294 views

What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?

For each genus $g$, there are many curves of genus $g$ defined over $\mathbb Q$. How many? We might study this question by considering the rational points of the Deligne-Mumford moduli space of curves ...
9
votes
1answer
554 views

Questions about the “universal elliptic curve” over the affine $j$-line punctured at 0 and 1728

So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not ...
125
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0answers
6k views

Grothendieck -sad news [closed]

Sorry for that this is not a real question. But I thought people would like to know. Alexandre Grothendieck died today: ...
5
votes
1answer
429 views

Disjoint images of polynomials

Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
1
vote
1answer
108 views

Normal surface is Cohen-Macaulay - reference

in this question http://mathoverflow.net/a/55528/61732 it is stated that a normal variety is CM outside a set of codim at least 3. That would imply that normal surfaces are CM. [edited:] I wanted to ...
1
vote
1answer
121 views

On holomorphic vector bundles over compact Kahler surfaces

Let $E\to X$ be a complex vector bundle over a compact Kahler surface $X$. Assume $c_{i}(E)\in H^{i,i}(X)$ for all i. Does the bundle $E$ admit a holomorphic structure?
4
votes
0answers
141 views

How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]

Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...
3
votes
1answer
117 views

Algebraization isomorphism, formal existence, mod p

Let $X$ be a smooth projective variety over a scheme $S$ being the spectrum of a discrete valuation ring of mixed characteristic $(0,p)$. Let $X_n$ be the respective thickenings of the reduced special ...
1
vote
0answers
44 views

Name for generalization of bivariate weighted-homogeneous polynomials

A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that ...
7
votes
1answer
265 views

Białynicki-Birula theory for non-complete varieties

I would like to know to which extent the theory developed for smooth projective varieties in the following articles A. Białynicki-Birula, Some theorems on actions of algebraic groups. Ann. of ...
1
vote
0answers
73 views

Does some square of the first Chern class preserved by conifold transition?

Let $X$ be a smooth projective 3-fold or a symplectic 6-manifold. Suppose $Y$ is a conifold transition on a single nullhomologous Lagrangian sphere $S^{3}$ in $X$. Then there is a exact sequence $0\to ...
7
votes
3answers
406 views

Gerbes and Stacks

The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, ...
4
votes
0answers
92 views

Real structure in the mixed Hodge structure associated to an isolated singularity

We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface ...
5
votes
1answer
160 views

Optimal definition of “paving by affine spaces”?

Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine ...
7
votes
1answer
207 views

Meaning of fibration in Kazhdan and Lusztig's paper on affine flag manifolds

Kazhdan and Lusztig's paper "Fixed point varieties on affine flag manifolds" has the following definition on p.143: define inductively a variety $Z$ to be an "almost affine space" if $Z$ is affine or ...
0
votes
0answers
57 views

Isomorphy of finite $R$-algebras under special conditions

Let $R=k[x_1, \ldots, x_n]$ be the polynomial ring over a field of characteristic zero. Let $R \subseteq S$ be a finite ring extension i.e. $S$ is finitely generated as $R$-module (free if that helps) ...
2
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0answers
85 views

Hodge Bundles on Tropical Spaces

I am not sure that this question even makes sense, which I suppose is part of the questions itself. In any case, I attended a talk recently wherin there was some discussion about a "tropical ...
3
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0answers
138 views

A Diophantine equation revisited

No integer solution of this Diophantine equation $$x^4+y^4+1=z^2$$ is known other than the trivial ones. While I was reading a paper of Don Zagier, I realized that his idea on the Euler's sum of ...
14
votes
2answers
894 views

Images of polynomials

Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all ...
1
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0answers
88 views

Projective tangent cones, ordinary singularities and blow-ups

Let $X\subset\mathbb{P}^n$ be a projective variety and let $Y\subset X$ be the singular locus of $X$. Assume that $Y$ is smooth. I would like to know if the following are equivalent: $X$ has an ...
14
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0answers
283 views

Does every relative curve have a Picard scheme?

More precisely: Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers are integral curves of genus $g$. Must the fppf relative Picard functor $\operatorname{\bf ...
13
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0answers
330 views

Etale fundamental group of a curve in characteristic $p$

Let $C$ be a connected, smooth, proper curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\pi_1(C)$ be the etale fundamental group of $C$ - I only care about ...
2
votes
1answer
124 views

Characteristic Varieties and Associated Varieties

Two notions that occur often in representation theory seem to be that of a "characteristic variety" and that of an "associated variety". The former term seems exclusive to D-module theory while the ...
1
vote
1answer
107 views

On the infinitesimal lifting property of non-singular affine schemes

Let $k$ be an algebraically closed field (not necessarily of characteristic $0$), $X$ a non-singular affine closed subscheme in $\mathbb{A}^n_k$ for some $n \ge 2$. Denote by $I_X$ the ideal of $X$ in ...
1
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0answers
61 views

vector space of ternary forms with real rooted property

Let $V \subseteq \mathbb{R}[x,y]_d$ be a two dimensional linear subspace of the vector space of bivariate forms of degree $d$. For each degree $d$ we can find such subspaces with the property that ...
4
votes
0answers
92 views

Existence of a torus fibration with given vanishing cycles

Suppose I have a torus fibration over the disc with $n$ nodal singular fibers $F_1,\dots,F_n$ over the points $p_1,\dots,p_n$. I was specifically thinking about a Lagrangian fibration, but I'd be ...
2
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0answers
65 views

A dual notion to Lawvere-Tierney operators for geometric surjections?

A geometric embedding into a Grothendieck topos can be characterised by giving the Lawvere-Tierney topology that induces it. This lets us reduce questions about subtoposes to more elementary ...
1
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1answer
116 views

Run MMP between varieties of isomorphic in codimension 1

Let $X, Y$ be two birational projective varieties which are isomorphic in codimension 1. Suppose $H$ is an ample divisor on $Y$, and $H'$ be its strict transform on $X$, suppose we can run MMP with ...
0
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0answers
124 views

a question about Deligne, Mostow's paper

I'm reading Deligne and Mostow's 1986 Publ.Math.IHES paper, entitled "Monodromy of hypergeometric functions and non-lattice integral Monodromy" I need to more details on page 72, more precisely, ...
6
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0answers
94 views

Etale local isomorphism to the tangent cone

Let $X$ be a scheme and $p\in X$ a closed point. We say that $(X,p)$ is etale locally isomorphic to $(Y,q)$ if there exists an etale neighborhood of $p$ in $X$, and etale neighborhood of $q$ in $Y$, ...
1
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0answers
86 views

Pre- and post-multiplication by diagonal matrices [closed]

Let $\mathbf{1}$ denote an $n\times 1$ vector with all entries equal to 1. Given an $n\times n$ matrix A with strictly positive entries, and non-negative diagonal matrices $D_1$ and $D_2,$ evaluate ...
7
votes
1answer
243 views

Embedding proper algebraic spaces

Does every proper algebraic space (over a field, say) admit a closed immersion into a smooth proper algebraic space? Remark: Of course, if we say "projective" instead of "proper" then the answer is ...
16
votes
3answers
902 views

How mirror of quintic was originally found?

In the 90-91 pager "A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY", Candelas, De La Ossal, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically ...
1
vote
2answers
179 views

Sheaf cohomology on non paracompact topological spaces

I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me. My reference is Godement's book "Topologie algebrique et theorie dex ...
2
votes
0answers
58 views

Abel-Prym map for Prym-Tyurin varieties

Let $(J,\Theta)$ be the Jacobian of a smooth projective curve $C$, and let $i:P\hookrightarrow J$ be an abelian subvariety of $J$ such that $i^*\Theta\equiv e\Xi$ for some principal polarization $\Xi$ ...
2
votes
0answers
141 views

computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]

How can we show that $$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])= \prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$ where $\operatorname{char}_T V$ denotes the character ...
1
vote
1answer
122 views

Are generically trivial finite unramified morphisms trivial

Let $S$ be a smooth affine variety over $\mathbb C$ and let $f:X\to S$ be a finite unramified morphism. Suppose that $X(K(S))$ is non-empty. (This means that $X\to S$ has a section generically. It ...
8
votes
0answers
343 views

How much algebraic geometry do I need to study complex geometry?

As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...
9
votes
2answers
438 views

When does a cubic surface pass through five lines?

The set of 5-tuples of lines in $\mathbf{P}^3$ is parametrized by the 20-dimensional product of Grassmannians $G(2,4)^{\times 5}$. The set of cubic surfaces is parametrized by a 19-dimensional ...
3
votes
1answer
172 views

Obtaining non-normal varieties by pushout

In his answer to this MO question, Karl Schwede claimed that every non-normal variety can be obtained by an appropriate pushout diagram, as sketched in that answer. This would give substance to the ...
5
votes
0answers
167 views

Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective

$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
10
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0answers
246 views

Coarse moduli spaces of stacks for which every atlas is a scheme

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group ...
5
votes
0answers
196 views

When did “Betti cohomology” come to be used the way it is today? (and how is it used)

This is sort of a mixture of a math and history question. First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I ...