Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,607
questions
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Variety of factorizations of differential operator
Take differential operator as polynomial of letter $d$ with coefficients in some function field, where $d$ act by derivation in this function field. Call it a differential field. For simplicity let ...
3
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2
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492
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isogeny and congruence subgroup
Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By ...
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1
answer
307
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Smoothness of the quotient surface by an involution with nice fixed locus
Let $X$ be a (smooth complex algebraic) surface. Suppose $\theta$ is an automorphism of order $2$ of $X$, such that its fixed locus is a disjoint union of smooth curves. I am trying to prove that the ...
2
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1
answer
200
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Integral values of rational map
This question is related to this post on Math.MO.
A theorem of B.Segre tells us that if there is one rational point on a non-singular cubic surface $X$ over $\mathbb{Q}$, then the surface is ...
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0
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372
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Hypersurfaces with Picard group generated by classes of lines on the same plane
For which values of $d$ is the following possible: There exist a smooth hypersurface $X$ in $\mathbb{P}^3$ of degree $d$ with Picard number $d$, containing $d-1$ lines $l_1,...,l_{d-1}$ on the same ...
1
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1
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513
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Minimal polynomial of symmetric endomorphism on abelian variety
Let $(A,\Theta)$ be a principally polarized abelian variety over an algebraically closed field $k$, and let $f$ be a symmetric endomorphism of $A$ (that is, $f^\dagger=f$ where $\dagger$ denotes the ...
3
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2
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360
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Kahler differentials on cluster varieties
On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
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Does Noether normalization hold more general? [duplicate]
Noether normalization tells us that a finitely generated $k$-algebra is an integral extension of a polynomial algebra over the field $k$.
My question is whether this still holds if we replace the ...
8
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3k
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Orbits of group scheme action
I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...
0
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1
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172
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$I/N$ is finitely presented module
Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$.
...
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Finding stable ideals of $\mathbb{F}_3[[X,S]]$ by group action
Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by:
$\sigma_k: X \mapsto X + S + X^k$
$\sigma_k: S \mapsto S + S^3$.
Conjecture: There exists a ...
4
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Computing fundamental groups of the complement of plane curves
This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an ...
3
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2
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897
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Gross's paper on Heegner points
I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points:
Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...
6
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0
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336
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Equivalence of Versions of the Affine Grassmannian
Let $G$ be a compact connected semisimple Lie group. The algebro-geometric definition of the affine Grassmannian is the coset space $$\mathcal{G}r=G_{\mathbb{C}}(\mathcal{\mathbb{C}((t))}/G_{\mathbb{C}...
25
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5
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is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?
This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
4
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What does it mean for a Deligne-Mumford stack to have trivial generic stabilizers?
I have stumbled upon some literature on Deligne-Mumford stacks, and it seems to me, at least superficially, that there is a strong link between DM-stacks which have "trivial generic stabilizers" and "...
3
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2
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801
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Fourier--Mukai transforms and adjunction
If $X$ and $Y$ are smooth projective varieties, $p: X \times Y \to X$ and $q: X \times Y \to Y$ are the projections, and $\mathcal{P}$ is an object in $D^b(X \times Y)$, then the Fourier--Mukai ...
0
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1
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477
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support of tor-sheaf
suppose $D$ is an effective smooth irreducible divisor on a smooth variety $X$.
suppose $W$ is a closed subvariety of $X$ not contained in $D$.
Suppose $L$ is a line bundle on $X$ and consider the ...
12
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2
answers
599
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The solutions of a system of polynomials
Given positive integers $m_1,...,m_n$, is it possible to solve the following equation system over the field of complex numbers?
$$m_1x_1+\cdots+m_nx_n=0$$
$$m_1x_1^2+\cdots+m_nx_n^2=0$$
$$\cdots$$
$$...
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242
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Restriction of relative sheaf on a section
suppose $X\rightarrow Y$ is a projective bundle or rank $r$.
Let $S$ be a section of this bundle.
What is the restriction of the relative tangent bundle $T_{X/Y}$ on the section $S$ ? Is it free ?
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Branch loci of Ramified covers
Let $X\to Y$ be a $d:1$ ramified covering map from a smooth complex projective variety $X$ to a smooth complex projective variety $Y$.
Question 1. What does the smoothness imply on the branch locus ...
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Chow groups and short exact sequence
Let $X$ and $Y$ be subvarieties of a smooth projective variety $M$ such that $M=X \bigcup Y$. Can you explain to me why $ A_k ( X \bigcap Y ) \to A_k ( X ) \oplus A_k ( Y ) \to A_k ( X \bigcup Y ) \to ...
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278
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Group of Hodge classes
I would like to know if there exists a relation between $ \mathrm{Hdg}_k(X \bigcup Y ) $, $ \mathrm{Hdg}_k(X) $, $ \mathrm{Hdg}_k( Y) $ and $ \mathrm{Hdg}_k(X \bigcap Y )$ ( short exact sequence, or ...
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0
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153
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Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber
Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
1
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0
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424
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Kernel of a multiplication map of global sections of line bundles
Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $C$ be a smooth projectively normal curve in $X$ and $\mathcal{L}$ be a line bundle on $X$ of the form $\mathcal{O}_X(-D)$ for some (...
3
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0
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157
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Monodromy along strata of a pushforward
Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived.
Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...
2
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0
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90
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Surjectivity of modules and Castelnuovo-Mumford regularity
Let $X$ be a smooth projective surface in $\mathbb{P}^t$ for some $t \ge 3$, $C$ a smooth curve in $X$ and $\mathcal{L}$ is a line bundle on $X$. Denote by $A_X$ (resp. $A_C$) the homogeneous ...
3
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0
answers
444
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Is the Picard number bounded by $b_2$ in positive characteristic?
We know that for a smooth projective variety $X$ over an algebraically closed field of characteristic 0 (for example $k=\mathbb{C}$), $\rho(X)\leq b_2(X)$. What about in positive characteristic? Is ...
12
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0
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570
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Cohomology and conifold transition for the quintic
Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$
it has 125 singular points whose links are homeomorphic to $S^2\times S^...
2
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2
answers
652
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Does $\Gamma_*$ commute with tensor product?
Given a coherent sheaf $\mathcal{F}$ we denote by $\Gamma_*(\mathcal{F})=\oplus H^0(\mathcal{F}(d))$. Suppose, $\mathcal{F}_1$ and $\mathcal{F}_2$ are two coherent sheaves on $\mathbb{P}^n$. Denote by ...
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1
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155
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integration along fibres and kunneth
suppose $X\times I$ is a product of a smooth manifold and unit interval.
There is a map $pi_*:\Omega^k_{X\times I} \rightarrow \Omega^{k-1}_X$ called integration along fibres.
Similar operation ...
3
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2
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384
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How should I think about this scheme constructed from a line bundle
Let $X$ be a scheme and $ \mathscr{L}$ be a line bundle on $X$. In a few proofs I have seen the scheme
$$ L = \mathscr{S}{\rm pec} \oplus_{n \in \mathbb{Z}} \mathscr{L}^{\otimes n} \to X$$
pop up. ...
6
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0
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525
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A question on Castelnuovo-Mumford regularity
Consider a short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ of coherent sheaves on $\mathbb{P}^n$. Assume that $\mathcal{F}''$ (resp. $\mathcal{F}$) is $m-1$ (resp. $m$...
13
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0
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871
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Stack of Tannakian categories? Galois descent?
I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\...
3
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1
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2k
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rational map becomes a morphism after blow-ups
In dimension 2, a rational map becomes a morphism after a sequence of blow-ups. Does this still hold in higher dimensions?
12
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1
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534
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Square of primary ideals
Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
3
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2
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2k
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homotopy exact sequence for the étale fundamental group
I have been trying to understand the homotopy exact sequence for the étale fundamental group which says
$$ 1 \rightarrow \pi_1 (\bar{X},\bar{x_0})\rightarrow \pi_1 (X,x_0)\rightarrow Gal(k)\...
4
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1
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467
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Kodaira classification and the McKay correspondence
Kodaira's table of singular fibers has a singular fiber
for each of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$; these are chains or cycles of (-2)-curves connected ...
11
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1
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668
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Do quantum "Sure-Shor separators" have a natural Veronese/Segre classification? (question inspired by Gil Kalai and Aram Harrow)
Aram Harrow asked: "Is there any place this is written up?"
Update Partly in answer to Aram's question, the thermodynamical properties of varietal dynamical systems now are written-up in ...
1
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1
answer
484
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Does flatness/smoothness over special fiber imply flatness/smoothness globally?
Let $f:X \to \mbox{Spec } R$ be a projective morphism between irreducible Noetherian schemes. Assume that $R$ is a discrete valuation ring and its residue field is algebraically closed. Suppose now ...
8
votes
1
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280
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Do level sets always correspond to even graphs?
Suppose I have a level set of some function $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^m$, say $L:=\{x:f(x)=c\}$. Let $S$ denote the points in $L$ at which $L$ is locally diffeomorphic to an open ...
15
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1
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2k
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Moduli space of motives vs moduli space of varieties
A (projective) abelian variety $A$ over the complex numbers is determined by $H^1(A,\mathbb{Z})$ together with its Hodge structure and polarization. This miracle means that one can parametrise ...
0
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2
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757
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Picard number of abelian variety [closed]
I would like references or a result about the computation of the picard number of the jacobian of an algebraic curve.
What about the special case when the picard number of the Jacobian is one (is ...
2
votes
2
answers
1k
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Degree of a smooth projective variety
Let $i_1:X \hookrightarrow \mathbb{P}^n$ and $i_2:Y \hookrightarrow \mathbb{P}^N$ be two projective schemes.
Let $f:X \to Y$ be a surjective projective morphism between smooth projective varieties ...
1
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2
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875
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Connections on the Hodge bundle?
Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...
3
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0
answers
448
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Deformation of a family of curves in a surface
Let $B$ parametrize a family of (reduced) curves in $\mathbb{P}^3$. Assume there exists a smooth hypersurface $X$. in $\mathbb{P}^3$ containing all the curves parametrized by $B$. In otherwords, for ...
7
votes
2
answers
1k
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Strata of the Affine Grassmannian
Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...
2
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2
answers
1k
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Uniqueness on square root of complex Line Bundle
Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
1
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1
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427
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Existence of rational section to a flat projective morphism
Let $X$ be a smooth projective variety over an algebraically closed field $K$ of dimension greater than $1$. Suppose there exists a flat projective morphism $f:X \to \mathbb{P}^n$ for some $n \ge 1$. ...
1
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1
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212
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multisections of grassmannian bundle
Suppose $\pi:G_X(r,N)\rightarrow X$ is a Grassmannian bundle associated to a vector bundle $E\rightarrow X$. Is it possible to construct a smooth/normal multisection
of $\pi$. Ie. is there a finite ...