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

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13 views

If the quotient of an algebraic space $X$ by a finite group is a scheme, is $X$ a scheme?

If the quotient of an algebraic space $X$ by a finite group $G$ is a scheme, is $X$ already a scheme? Here $G$ is just a finite group, but I'd like to know the answer when $X$ is defined over ...
0
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0answers
11 views

A geometric construction of the complex projective plane?

The paper Kötter's synthetic geometry of algebraic curves, (N. Fraser, Proceedings of the Edinburgh Mathematical Society 7, 46–61, 1888) opens with a sketch of what appears to be a synthetic ...
2
votes
2answers
168 views

Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?

Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...
0
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1answer
52 views

Extending a section of a coherent sheaf and homomorphism

Let $X=\mathrm{Spec}(A)$ be an affine integral scheme of finite type over $\mathbb{C}$ and $\phi:\mathcal{F} \to \mathcal{G}$ be a surjective morphism of coherent sheaves on $X$. Let $f \in A$, ...
1
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0answers
55 views

Generators for fake projective planes groups

Is there a reference for generators of fundamental groups of (some) fake projective planes in terms of matrices in $SU(2,1)$?
2
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1answer
338 views

Is the upper half plane an algebraic stack?

Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$. So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli ...
9
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1answer
220 views

Higher Fano varieties and Tsen's theorem

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...
4
votes
1answer
693 views

Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...
2
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1answer
188 views

A generalization of miracle flatness theorem

I wonder if the miracle flatness theorem Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension still works if the rings involved are not local (and the dimension condition is ...
12
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430 views

Meaningful review of Moriwaki's “Arakelov Geometry”

I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry book: http://www.ams.org/bookstore-getitem/item=mmono-244 I could do the review the standard way in a day or ...
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37 views

Coefficients of the pull-backs of divisors by resolving morphism

Let $\varphi : X \dashrightarrow X$ be a rational map. By a theorem of Hironaka we can find a resolution of singularities $(\tilde{X}_\varphi,\pi)$ of $\varphi$, where $\tilde{X}_\varphi$ is a ...
0
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1answer
401 views

What is the meaning of non-Hausdorff spaces in algebraic geometry [closed]

At the beginning I should warn everybody reading this post: I don't know much about algebraic geometry so specialists in this subject may see my question as ignorant. As far I understood one on the ...
1
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1answer
167 views

symplectic reduction for pair $(M,D)$

Let $M$ be a symplectic manifold with divisor $D$. Then how can we define symplectic reduction for pair $(M,D)$?
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0answers
146 views

Why the geometry of pair $(X,D)$ is important [on hold]

Let $(X,D)$ be a pair where $X$ is an algebraic variety and $D$ is a divisor on $X$. Why the geometry of pair $(X,D)$ is important?
2
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0answers
100 views

A multivariable polynomial degree question

Given $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$, minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$, $$\mathsf{F(p)}=a, ...
1
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0answers
199 views

Algebraicity of the stack of coherent sheaves

I am trying to understand the proof of Theorem 4.6.2.1 in the book on algebraic stacks by Laumon and Moret-Bailly. The setting is this: $S$ is a Noetherian scheme, $f\colon X \rightarrow S$ is a ...
1
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0answers
97 views

Characterizations of regular holonomic D-modules

I'm looking for references for the various characterizations of regular holonomic D-modules, in particular proofs of their equivalence. For instance, some characterizations I've seen (in the analytic ...
1
vote
1answer
142 views

Dimension of Ext modules [on hold]

Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
2
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1answer
139 views

Second cohomology groups of Nakajima quiver varieties

Let $X=M(v,w)$ be a Nakajima quiver variety for a quiver $Q$. Can one calculate the second singular cohomology groups $H^2(X,\mathbb Z)$ or $H^2(X,\mathbb C)$ explicitly, and if not, are there some ...
0
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1answer
157 views

Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...
6
votes
2answers
561 views

fundamental groups of curves

I saw the following statement made without proof in a paper of Bogomolov and Tschinkel: If $X$ is an algebraic surface, and $C$ is an ample smooth curve in $X,$ then the fundamental group of $C$ ...
1
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1answer
82 views

Induced topology on site + Reconstructing global sections of a scheme (Orlov)

Let $(C,T,O)$ be a ringed site. Let $X$ be a presheaf on C. We get an induced ringed site $(C/X,T_X,O_X)$. C/X is the over category wrt the presheaf X. The topology $T_X$ is the biggest topology ...
0
votes
1answer
96 views

Degree of the negative part of a divisor

Let $K$ be an algebraically closed field (or $\overline{\mathbb{C}(z)}$ for a more precise condition). And let $P \in K[x,y]$ be an irreducible polynomial of degree $m$ with respect to $x$ and degree ...
6
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0answers
225 views

What is an excellent algebraic space?

What does it mean to say that an algebraic space $S$ is excellent? One knows that excellence of a Noetherian ring is not a property that is etale local (that is, excellence cannot be checked over an ...
2
votes
0answers
172 views

Algebraic closedness in field of fractions

If $A\subseteq B$ are affine domains over an algebraically closed field $k$ of characteristic zero, such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
2
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0answers
106 views

Do we have the following “devissage commutative diagram” in K-theory?

Let $X$ be a non-reduced Noetherian scheme. We define $K^0(X)$ to be the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ to be the Grothendieck group of the derived category ...
9
votes
0answers
128 views

Why are unramified maps not required to be locally of finite presentation?

I have read and heard several times that it is “important” that unramified maps are not required to be locally of finite presentation, but only locally of finite type. Apart from this issue with ...
0
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0answers
175 views

Complete Intersection

Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of ...
0
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0answers
78 views

The injection of direct image sheaf

Let $f:X \longrightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds and $L$ be a holomorphic line bundle on $X$. Let $m$ be a positive integer. We denote by ...
3
votes
0answers
91 views

Action of automorphisms on cohomology with supports

Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
5
votes
1answer
150 views

Existence and uniqueness of extensions of a finite flat map

Suppose that $S$ is smooth and that $U\subset S$ is a dense open subscheme. Let $X$ be a scheme (not necessarily smooth) and let $f:X\to U$ be a finite flat morphism. I would like to know whether ...
21
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1answer
1k views

Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...
9
votes
1answer
316 views

Can every algebraic variety of dimension $n$ be covered by $n+1$ affine opens?

Suppose $X$ is a complete algebraic variety of dimension $n$. Must there exist an affine covering with $n+1$ pieces? (For a projective variety in $\mathbf{P}^m$, we can always project it to some ...
4
votes
1answer
532 views

Recent ideas in Macaulayfication?

Kawasaki has shown that a quasi-projective variety over a field has a Macaulayfication. His construction does not preserve the Cohen-Macaulay locus of the original variety, only a finite set of ...
0
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1answer
72 views

A question of direct image of relative canonical bundle

Proposition: Let $f:X\rightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds, and $L$ be a holomorphic line bundle. Then there exists a Zariski open set $Y_0\subset Y$ such that ...
0
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0answers
51 views

Is a general extension of general stable sheaves on $\mathbb P^2$ stable?

Theorem 2 in this paper by Bhosle gives a nice condition on slopes for when a general extension of general stable bundles on curves is stable. Does anyone know whether there is an analogous result for ...
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54 views

Degree of function field extension in several variables (degree of an endomorphism over an AV)

I just want to know which is the best way to calculate the degree of a function field extension like this $[\mathbb{F}_q(a,b,c):\mathbb{F}_q(x,y,z)]$ where $x\mapsto f(a,b,c)$ $y\mapsto g(a,b,c)$ ...
0
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0answers
149 views

Determine existence of irreducible variety in given homology class

Given a homology class $\alpha \in H_k(X,\mathbb{Z})$ on a variety $X$, is there a way to determine if there exists an irreducible subvariety $Y \subset X$ that has that class, i.e. $[Y] = \alpha$? ...
25
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1answer
873 views

Combinatorics of K(Z,2)?

Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...
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0answers
114 views

Must read books on topics in IMO [on hold]

I know there was this post Good books on problem solving / math olympiad However, I was looking for books that don't just approach in problem solving, but talk about the development of the theory. ...
3
votes
2answers
187 views

What are the easiest examples of irreducible, but not big, monodromy representations

Let $\rho: \pi_1(S,s_0) \to GL(V)$ be the monodromy representation associated to a local system of $\mathbb Q$-modules $\mathbb V$ with $\mathbb V_{s_0} = V$. Let $H$ be the Zariski closure of the ...
2
votes
1answer
359 views

Two basic questions concerning geometrically ruled surfaces. [on hold]

Good Morning, These questions are a result of trying to understand the proof of Proposition III.18 in Beauville's book 'Complex Algebraic Surfaces'. Here is the setup - everything is smooth, ...
5
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2answers
267 views

References for the moduli space of complex structures

I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
6
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2answers
296 views

Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
1
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1answer
72 views

Multiplicity of the intersection of a Rational curve in a quadric with a tangent plane

Consider a rational map $u : \mathbb{CP}^1 \to \mathbb{CP}^4$ of degree~$d$, such that the image lies in a fixed 3-dimensional quadric $Q^3$. In other words, its image is a rational curve in $Q^3 ...
2
votes
1answer
158 views

Automorphisms of complete local rings

Let $k$ be a field and $(A,m)$ be the completion of the local ring of a smooth point of a $k$-variety. Let $x_1,x_2\in m\backslash m^2$ be regular elements. I am interested in knowing if one can find ...
1
vote
2answers
161 views

Examples of quotients by infinitesimal group schemes

I'm looking for examples of explicit actions of the infinitesimal group schemes $\alpha_{p^n}$ on schemes (maybe as simple as the affine plane) in characteristic $p$ or mixed characteristic, and their ...
12
votes
4answers
3k views

A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite

I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/ ...
25
votes
3answers
1k views

Reverse mathematics of (co)homology?

Background Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 ...
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1answer
338 views

Atiyah's vector bundles over an elliptic curve

I'm reading through Atiyah's paper that classifies vector bundles over an elliptic curve, and I'm a little confused about one of his proofs. Lemma 15(i) states that if $E \in \mathcal{E}(r,d)$ is a ...