# Tagged Questions

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

**2**

votes

**0**answers

22 views

### A question on Selmer groups

Let $p $ be a nonarchimedian prime of $\mathbb {Q} $, $K$ a number field, $\nu$ a nonarchimedian prime of $\mathbb {Q} $ that remains inert in $K$ and is different from $p $.
Denote by $L$ the ...

**4**

votes

**0**answers

27 views

### Infinitesimal deformations of the formal group of $\mathbb{G}_m$

For a commutative ring $R$, consider the formal group $\widehat{\mathbb{G}}_m$ over $R$ that is the completion of $\mathbb{G}_{m, R}$ along its identity section (naively, $\widehat{\mathbb{G}}_m$ is ...

**3**

votes

**1**answer

60 views

### Quasicoherent sheaves on superschemes

I am interested in learning about super algebraic geometry (some objects I am studying seem to be naturally superstacks, at least in some sense). What would be the best reference for the subject? I am ...

**0**

votes

**0**answers

59 views

### Classical cohomology versus Zariski cohomology

I have two questions on the computation of the zero cohomology group with integer coefficients of a closed subset of $\mathbb{C}^{n}.$ This problem is related to a hyperplane arrangement problem I am ...

**0**

votes

**0**answers

18 views

### How to get an explicit deformation which splits the support of a finite length quotient?

Assume $X$ is a smooth projective surface over say $\mathbb{C}$, $E$ is locally free sheaf of rank $r=2n$ on $X$ and $p\in X$ is a fixed point.
Let $E\rightarrow Q$ be a finite length quotient, such ...

**0**

votes

**0**answers

90 views

### cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian.
Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where ...

**0**

votes

**0**answers

46 views

### Proximal point avoids a subvariety

I have a seemingly easy problem about the proximal point in a variety of a general point in the ambient space, which I don't have a proof:
Let $X \subset \mathbb{C}^N$ be the affine cone over some ...

**3**

votes

**0**answers

135 views

### Example of a genus-1 degree-7 plane curve

I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1.
One can show that for a general ...

**6**

votes

**3**answers

304 views

### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

**3**

votes

**1**answer

196 views

### Are essentially smooth schemes noetherian?

Let $k$ be a field. I am unable to find a precise definition of essentially smooth $k$ schemes, but I will stick to this definition below, since this is exactly what I need:
Definition: A $k$-scheme ...

**1**

vote

**0**answers

52 views

### are extensions of flat connections flat? [migrated]

Let $U$ be a smooth complex variety and $X$ a compactification by a normal crossings divisor $D$. Let $E$ be a vector bundle on $U$ (i.e. locally free $\mathcal{O}_U$-module), together with a ...

**1**

vote

**0**answers

96 views

### a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let ...

**5**

votes

**1**answer

175 views

### Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ ...

**2**

votes

**0**answers

222 views

### How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?

This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments.
Let $X$ be an algebraic variety over ...

**5**

votes

**0**answers

107 views

### Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?

According to the Keel-Mori theorem if $\mathcal{X}$ is a separated Artin stack of finite type over a scheme S with finite inertia, then it admits a coarse moduli space $X$ (which is an algebraic ...

**-4**

votes

**0**answers

131 views

### About Hodge conjecture [on hold]

Good evening everyone,
Can someone inform me about the current state of research in the field of the Hodge conjecture ? Where are we currently stops in mathematical research in this area?
Thank you ...

**0**

votes

**0**answers

108 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 ...

**7**

votes

**1**answer

225 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 ...

**0**

votes

**1**answer

79 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 $f \in A$, ...

**2**

votes

**0**answers

76 views

+100

### 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 $SU(2,1)$?

**12**

votes

**0**answers

501 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 ...

**2**

votes

**1**answer

202 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 ...

**-1**

votes

**0**answers

41 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 ...

**4**

votes

**1**answer

723 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 ...

**-1**

votes

**0**answers

148 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**

votes

**0**answers

104 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**

vote

**0**answers

102 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

**1**answer

149 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**

votes

**1**answer

141 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 ...

**1**

vote

**0**answers

209 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 ...

**9**

votes

**1**answer

239 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 ...

**1**

vote

**1**answer

187 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 $(M,D)$?

**6**

votes

**0**answers

238 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 ...

**1**

vote

**1**answer

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 ...

**2**

votes

**0**answers

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

**0**answers

131 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**

votes

**1**answer

113 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 ...

**2**

votes

**2**answers

191 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**

votes

**0**answers

52 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 ...

**5**

votes

**1**answer

163 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 ...

**0**

votes

**1**answer

74 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 ...

**9**

votes

**1**answer

325 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 ...

**-1**

votes

**0**answers

55 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
$x\mapsto f(a,b,c)$
$y\mapsto g(a,b,c)$
...

**3**

votes

**0**answers

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 ...

**0**

votes

**0**answers

178 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
...

**3**

votes

**2**answers

189 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 ...

**5**

votes

**2**answers

271 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 ...

**1**

vote

**2**answers

162 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 ...

**2**

votes

**1**answer

159 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

**1**answer

73 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 ...