# Tagged Questions

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

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### What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...

**2**

votes

**1**answer

27 views

### Simple example of isolated critical point with non-semisimple monodromy

Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...

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43 views

### Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...

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**2**answers

116 views

### Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let
$$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$
be a non-singular ...

**6**

votes

**1**answer

139 views

### The moduli space of special Lagrangian submanifolds

Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures on the base space $B$. A theorem of Hitchin says that there ...

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**1**answer

110 views

### Blow-up of $\mathbb{P}^4$ along a quadric surface

Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...

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40 views

### Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...

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43 views

### plotting parametrized algebraic curves near singularities

I have a parametrized algebraic curve:
x(t)=A(t)/D(t);
y(t)=B(t)/D(t);
with A(t) and B(t) being polynomials in t. The curve is solution of a linear system in two unknowns x and y with Cramer's ...

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58 views

### Local system over $\mathcal A_{g,[n]}$ with unipotent monodromy

Let $\mathcal A_{g,[n]}$ denote the moduli space of principal polarized abelian varieties with level-[n] structure and
$\bar {\mathcal A}_{g,[n]}\supset \mathcal A_{g,[n]}$ a smooth Toroida ...

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72 views

### Degree and quasi projective family

Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$.
Question : $\exists ...

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67 views

### Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...

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**3**answers

1k views

### Algebra and Cancer Research

Let me start by acknowledging the existence of this thread: Mathematics and cancer research ?
It is well-known that mathematical modeling and computational biology are effective tools in cancer ...

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**2**answers

308 views

### Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$

This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...

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233 views

### What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...

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**1**answer

180 views

### Is the big cell a principal open set?

Let $G$ be a complex affine reductive algebraic group, $B\subseteq G$ a Borel with maximal torus $T$ and unipotent radical $U$. Let $w\in\operatorname N_G(T)$ be a representative of the longest Weyl ...

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**1**answer

197 views

### Fermat surface known to have very few rational integer solutions

The motivation for this question is the Selmer curve, given by
$$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$
One can show that this curve has no rational integer solutions, despite having a solution ...

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48 views

### Surjectivity of $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$

Let $R$ be a Noetherian ring and let $M$ is finitely generated $R$-module.Suppose $p$ is a minimal prime in $\text{Supp}_RM$. Then $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$ that $f(m)=m /1 $ is ...

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**1**answer

139 views

### Proof of the Belyi's theorem: where it is really used the hypothesis?

Consider the Belyi's theorem:
If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most ...

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**2**answers

303 views

### Do most degree $d$ morphisms of $P^n$ have smooth critical locus?

Let $f=[F,G,H]:\mathbb{P}^2\to\mathbb{P}^2$ be a morphism of degree $d\ge2$.
The critical locus $C_f$ of $f$ is the zero-locus of the Jacobian
determinant:
$$
C_f = \left\{ [x,y,z]\in\mathbb{P}^2 ...

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61 views

### Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.
Explicitly how would one go about computing ...

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**1**answer

151 views

### Hasse principle and Brauer-Manin obstruction for forms of large degree

The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the ...

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83 views

### Over which fields (of positive characteristic) is the Beilinson-Soulé vanishing conjecture known to hold?

Let $k$ be a field, and denote by $K_p(k)^{(n)}$ the weight $n$ eigenspace of the Adams operations on the $p$-th $K$-group of $k$.
The Beilinson-Soulé (BS) vanishing conjecture predicts that
$$
...

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**1**answer

139 views

### flat descent for perverse sheaves

Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...

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51 views

### Cohomology of Sym(E) [migrated]

Let $U,V$ be two vector bundles on some scheme $X$. Let $E$ be some bundle in $0\to U\to E\to V\to 0$. If I can show that $H^i(X; Sym(U\oplus V))=0$ for $i>0$, can we conclude the same property for ...

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**1**answer

138 views

### A covering lemma of Kawamata

In the paper "A generalization of Kodaira-Ramanujam's vanishing theorem", Kawamata states a covering lemma (Lemma 5) which is
Let $X$ be a non-singular projective variety, and $D$ be a divisor ...

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votes

**1**answer

109 views

### Quotient of product of curves

Let $C_1,C_2$ be smooth, projective curves of genera $g_1,g_2 \geq 2$. Assume that a group $G$ of order $(g_1 - 1)(g_2 - 1)$ acts on $C_1$ and $C_2$ such that $C_1/G \cong \mathbb{P}^1$ and $C_2/G ...

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94 views

### Can the property of essential finite type checked at a point?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian.
Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has ...

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**1**answer

194 views

### representation of algebraic fundamental group of projective line minus three point

everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...

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votes

**1**answer

168 views

### Canonical lifts from $\mathbb F_q$ and CM-theory

One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...

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**6**answers

910 views

### Algebraic Geometry for non-mathematician [closed]

I think I sound stupid but I have heard a lot about Algebraic Geometry as a subject and wish to study it without actually studying abstract algebra. I have never studied abstract algebra since I am a ...

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211 views

### Van den Bergh Duality, Serre Daulity and Poincaré duality [closed]

All three duality theorems:
Van den Bergh Duality, Serre Duality and Poincaré duality seem to be very similar, is there an explicit relationship between the three?
For example can van den Bergh ...

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146 views

### Surjectivity of $f\colon\Gamma_Z(M)\rightarrow\bigoplus_{p\in Z\backslash Z'}\Gamma_{pR_p}(M_p)$

Suppose $Z'\subseteq Z\subseteq\text{Spec} R$ such that every element in $Z\backslash Z'$ is a minimal element (with respect to inclusion as ideals) in $Z$. Assume further that both $Z$ and $Z'$ are ...

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116 views

+50

### Pullback of a sheaf associated to a divisor

I am reading a paper Desingularisation des varietes de Schubert generalisees by Demazure. I am interested in Lemma 3 on page 58. In particular, I would like to know whether the lemma is true and how ...

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**1**answer

223 views

### Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...

**3**

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**1**answer

143 views

### Smooth mixed hodge modules - representations of fundamental group?

I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...

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**0**answers

69 views

### degree of isogenies between Jacobians and Abelian Varieties

Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...

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**1**answer

269 views

### Proving that any two points on a variety can be joined by a curve; why does Bertini apply?

I want to prove the following statement:
For any two points $x$ and $y$ in an irreducible variety $X$, there is a one-dimensional, irreducible subvariety $C\subseteq X$ containing $x$ and $y$.
...

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votes

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302 views

### Existence of local sections

I would like to know when the property of "having a section" for a morphism of varieties in characteristic $0$ can be detected by spreading out to characteristic $p$.
Take a number field $K$, and let ...

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**1**answer

312 views

### why quintics are Calabi-Yau?

Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?

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113 views

### singularities and zeros of series and reverse problem [closed]

Given a series $$S=\sum_{n=1}^{\infty}a_n x^n,a_n\in \mathbb{N}$$ we know that it may have zeros,poles,branch points or natural boundary.Reversely,given it's zeros,poles,branch points or natural ...

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269 views

### Weak Fano and Log fano varieties

A projective smooth variety $X$ is weak Fano if $-K_X$ is nef and big. We say that $X$ is log Fano is there exists a divisor $D$ such that $-(K_X+D)$ is ample and $(X,D)$ is Kawamata log terminal.
Is ...

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**1**answer

183 views

### Surfaces singular along a curve

Let $C\subset\mathbb{P}^3$ be a smooth curve a degree $d$ and genus $g$. Let $\mathcal{S}$ be the system of surfaces of degree $k$ in $\mathbb{P}^3$ containing $C$ with multiplicity $\beta$.
What is ...

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**1**answer

341 views

### Two questions about the grassmannian

There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference.
...

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**1**answer

188 views

### Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, ...

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77 views

### Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...

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votes

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36 views

### connected components of a real algebraic variety and its hyperplane section

Let $X$ be a smooth projective variety of dimension at least $2$ over the real numbers $\mathbb{R}$ and $H \subset X$ a smooth hyperplane section. Assume that the set of real points is non-empty for ...

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**1**answer

336 views

### Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...

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**1**answer

324 views

### On a question motivating Lurie's treatment of formal moduli problems

Lurie, in his ICM 2010 proceedings paper Moduli Problems for Ring Spectra (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is ...

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**1**answer

120 views

### Universal coefficient theorem for local ring

Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat ...

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169 views

### When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space?

Let $p_1,...,p_k\in\mathbb{P}^2$ be general points. Let us consider the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$. It is clear that if $k\leq 3$ then $X_k$ is toric and hence a Mori Dream Space. ...