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

**3**

votes

**2**answers

285 views

### Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements.
Let $h,k\geq1$ be integer numbers and let ...

**2**

votes

**1**answer

159 views

### What does the Chern-Schwartz-MacPherson class of a singular variety look like?

Let $A_\ast$ and $F_\ast$ be the functors $\textrm{Var}_\mathbb C\to \textrm{Ab}$ of Chow groups and constructible functions, respectively, with respect to proper maps. Then the ...

**0**

votes

**0**answers

126 views

### Do there exist nontrivial motivic cohomology operations preserving weights?

Suppose that for each field $F$ a linear map $X(F): H_M^{p,q}(F, \mathbb{Q}) \longrightarrow H_M^{p,q}(F,\mathbb{Q})$ is given, such that $X$ commutes with inclusions of fields and transfers for ...

**1**

vote

**1**answer

195 views

### Composition of rational functions

Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, ...

**2**

votes

**0**answers

36 views

### A weaker version of Randell Isotopy Theorem

I am studying a problem in hyperplane arrangement theory related to the homotopy type of the complement manifold of a certain class of hyperplane arrangements.
In a well celebrated paper Richard ...

**0**

votes

**0**answers

65 views

### Why is the evaluation map of sheaves injective [migrated]

Let $E$ be a globally generated vector bundle of rank $r$. Let $V$ be a subspace of $H^0(S,E)$ of dimension $r$. We have the evaluation map $ev:V\otimes \mathcal{O}_S\longrightarrow E$. Why is this ...

**0**

votes

**0**answers

50 views

### Criterion for transverse boundary intersection of one-parameter family in $\overline M_{g,n}(X,\beta)$

Suppose we have a projective, nonsingular, convex variety $X$, $\beta \in H_2(X,\mathbb{Z})$ and a family
$$
\begin{array}{ccc}
\mathcal{C}& \to & X \cr
\downarrow& & & \cr
B ...

**1**

vote

**1**answer

135 views

### Chow groups of locally trivial affine fibrations

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$.
A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong ...

**14**

votes

**1**answer

636 views

### Do I know what “coherent sheaf” means if I know what it means on locally Noetherian schemes?

I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...

**0**

votes

**2**answers

95 views

### Example of a covariant derivative on a non-projective bundle

I was looking for a simple example of a covariant derivative on a bundle, where the bundle is not projective. If necessary, the example could be from complex or noncommutative geometry, but I would ...

**-2**

votes

**0**answers

69 views

### Cohomology of conic bundle 3-folds [migrated]

It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map ...

**1**

vote

**1**answer

166 views

### Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...

**4**

votes

**1**answer

98 views

### Lindel's theorem for semisimple simply connected G

Let $k$ be a field.
$G/k$ be a simply connected semisimple algebraic group.
Let $X/k$ be a smooth affine $k$-scheme.
Question: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back ...

**0**

votes

**0**answers

62 views

### Degree of permutation of hypercube

Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...

**0**

votes

**0**answers

48 views

### A multivariate polynomial question

Split $\{0,1\}^n$ into $S_0[n],S_1[n]$ with $S_0[n]\cup S_1[n]=\{0,1\}^n$ while $S_0[n]\cap S_1[n]=\emptyset$.
For every $n$, let $f(x_1,\dots,x_n),g(x_1,\dots,x_n)\in\Bbb R[x_1,\dots,x_n]$ be ...

**1**

vote

**0**answers

86 views

### On the local Euler obstruction for singular varieties

Let $X$ be a complex algebraic variety (not necessarily irreducible, nor reduced). Then the local Euler obstruction is a group isomorphism $$\textrm{Eu}: Z_\ast X\to F_\ast X,$$ where $Z_\ast X$ is ...

**1**

vote

**0**answers

116 views

### Are two line bundles with the same ramification type necessarily isomorphic?

I have no motivation for the following problem, I am just curious if it is true or not. Here it is:
If $l_1$ and $l_2$ are two complete $g^r_d$'s on a smooth curve $C$ such that the vanishing ...

**2**

votes

**0**answers

119 views

### When is the Hom-scheme connected?

Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor ...

**9**

votes

**2**answers

264 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

176 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

**1**answer

268 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 ...

**1**

vote

**0**answers

64 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 ...

**4**

votes

**0**answers

169 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 ...

**8**

votes

**3**answers

448 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

218 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

**1**answer

159 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

210 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
$$ ...

**3**

votes

**0**answers

265 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 ...

**12**

votes

**1**answer

246 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 ...

**0**

votes

**0**answers

118 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

245 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

89 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 $f \in A$, ...

**3**

votes

**0**answers

163 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 $SU(2,1)$?

**11**

votes

**0**answers

545 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

240 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

48 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

766 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**

votes

**0**answers

109 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

109 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

159 views

### Dimension of Ext modules [closed]

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

155 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 ...

**3**

votes

**1**answer

373 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

266 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

203 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 $(M,D)$?

**8**

votes

**1**answer

398 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

83 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

116 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 ...

**12**

votes

**1**answer

219 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

116 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 ...