All Questions
22,546 questions
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86
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Non log-canonical Fano of Picard number 1
What is an example of a Fano variety of Picard number 1 that does not have log-canonical singularities?
- 1
0
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0
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163
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Subtori of restriction of scalars
Let $E/F$ be a finite separable extension of a commutative field $F$. Let $T$ be the torus
${\rm Res}_{E/F}\; {\mathbb G}_m$, where ${\rm Res}$ is Weil's restriction of scalar. Is there a simple ...
- 6,349
1
vote
0
answers
113
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relation between $C \cdot K_S$ and $K^2_S$ for a curve $C$ on a complete intersection surface $S$
Let $S$ be a smooth complex projective surface which is a complete intersection and such that $K_S=\mathcal{O}_S(k)$, $k >0$.
Let $C$ be a smooth curve on $S$ such that $C^2 >0$.
I'm ...
- 135
3
votes
0
answers
242
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How to find special hermitian metrics on vector bundles
Given a regular scheme $X$, projective and flat over $Spec(\mathbb{Z})$, i.e. an arithmetic variety and a vector bundle $E$ on $X$. We get an associated vector bundle $F$ on the associated complex ...
- 1,391
3
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249
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Inverse limits of finite unramified covers, and the need for formally unramified covers
This question evolves mostly from my surprise at the answer to a previous question of mine (Is every flat unramified cover of quasi-projective curves profinite?). My surprise was at that inverse ...
- 1,522
4
votes
1
answer
275
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Comparing maps of reduced schemes
Nice fact:
Suppose f:X->Y is a map of schemes and Z⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and Z are reduced, then it follows that f factors through Z. ...
- 11.3k
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151
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An (almost) terminological question: could one shorten the phrase 'the spectrum of the residue field of a point'?
For a scheme S I want to consider the spectra of the residue fields of points of S. Is there any way to make this phrase shorter? Is there a term for the morphism that connects such a spectrum with S?
- 16.9k
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82
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Extending functions on curves to functions on abelian varieties
Say I have a function $f_g$ on the moduli space of curves $M_g$ for all $g\geq 1$. Is there some way of extending this to the moduli space of abelian varieties $A_g$ in a nice way?
What if I have ...
- 163
7
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0
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205
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sheaves on thickened nodal cubics
Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z]
is an irreducible cubic form defining a plane curve C with a node. A lot is known about
sheaves on C; for ...
- 5,422
1
vote
0
answers
169
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Partial ordering of vector bundles on projective spaces
I would like to know if there are some interesting partial orders defined on the isomorphism classes of vector bundles on $\mathbb P^n_k$ (you can assume $k$ is $\mathbb C$ if that helps).
...
- 30.6k
3
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175
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What is the difference between K-theory and K-cohomology of a stack with coefficients in Vect
In B.Toen's thesis, he defines two concepts: K-theory and K-Cohomology with coefficients in Vect. What is the difference between them? At best, is there some simple example, like weighted projectve ...
- 39
1
vote
0
answers
121
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Existence of Local Maxima in Connected Components
Question: For $f \in \mathbb{R}[x]$ is it true
that there is a local maximum of $g(x) = \frac{f(x)^2}{(x^2+1)^{d+1}}$ in each connected component of $f \neq 0$ where $d = \deg(f)$?
Comments: I haven'...
- 143
1
vote
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146
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Are spherical codes algebraic?
Jeffrey Wang in Section 4.2 writes "Since a code is the solution to a number of polynomial equalities between the shortest edges, the coordinates of each rigid point in the code are algebraic and lie ...
- 11
2
votes
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answers
160
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Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus
This question is related to my related post:
Volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map
The setting here is as follows:
Let $p: \mathbb{R}^{2k} \to \...
- 4,466
2
votes
0
answers
70
views
Associating an ideal to a subdivision
Given a coherent subdivison of a fan, how does one find a torus invariant ideal sheaf whose (normalized) blowup is the given subdivision?
4
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223
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Is there a sensible notion of abstract constructible space?
In the past by the term "variety" people understood a subset of projective space locally closed for the Zariski topology. Now we have a more natural notion of abstract algebraic variety, i.e. a scheme ...
- 23.4k
2
votes
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answers
190
views
Nilpotent derivations and geometric embeddings in mathematical physics
I am attempting to find my footing as an student who is ending his undergraduate studies and getting ready to embark on his journey to grad school in pure maths. In this capacity, I have been doing a ...
- 1,539
3
votes
0
answers
183
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morphisms on Quadric
Hello everyone,
A quick question, as I'm not sure I got it right.
Let $X=\mathbb{P}^1 \times \mathbb{P}^1$ and let $\mathcal{O}(a,b):=\pi^*_1\mathcal{O}(a)\otimes \pi^*_2\mathcal{O}(b)$. Is there a ...
1
vote
0
answers
165
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Simple questions about local parameters for a relative smooth curve endowed with a section
Let $f:X\longrightarrow S$ be a morphism of preschemes which is smooth of pure relative dimension 1. Let $\sigma:S\longrightarrow X$ be a section of $X$. Let $D$ be the (positive) divisor associated ...
- 411
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119
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Embeddings of spherical Kac-Moody varieties?
There is a well established theory of embeddings $G/H \to X$ into a normal variety $X$ where $G$ is a reductive group over an algebraically closed field and the image of any Borel $B$ is open in $X$; ...
- 3,968
1
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96
views
When the class of a complex is necessarily equi-dimensional
Let $P$ be a smooth projective variety. For an object $X$ of $D_{perf}(P)$ (i.e. a bounded perfect complex of $\mathfrak{O}_P$-module sheaves) we consider its class $[X]$ in $K_0(P)\otimes \mathbb{Q}(\...
- 16.9k
2
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1
answer
167
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Triviality of the Hodge bundle for a special family of semistable curves
Let g,h be positive integers. Let E be an elliptic curve, C be a genus h curve, and D be a genus g-h-1 curve. Let c,d,e be points on (resp.) C,D, and E.
Let f:CC --> E-e be the family whose fiber ...
- 10.5k
3
votes
0
answers
146
views
Comparing etale covers of a curve on two different algebraically closed fields $K \subseteq L$
Say that $K \subseteq L$ are two algebraically closed fields of characteristic $0$. Let $C$ be a curve (not nec. proj., but maybe assume smooth) over $K$. Supposedly $\pi_1(C_L)$ maps isomorphically ...
- 6,348
2
votes
0
answers
68
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Conditions for uniform limit of sets
This question roughly translates to seeing if the intersection of two limits of sets,
is the same as the limit of the intersections. This is of course false in general, but what conditions needs to be ...
- 15.8k
4
votes
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answers
167
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Components of variety of subalgebras
This question is motivated by the question Subalgebras of matrices and its answer by Mariano. We consider $X_{n,d}$, the variety of $d$-dimensional subalgebras not necessarily with 1 (with 1 makes ...
- 12.4k
1
vote
1
answer
162
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Does automatic decomposition of varieties into irreducibles exist?
Varieties decompose uniquely into finitely many irreducibles, and each variety is generated by only finitely polynomials. These two finiteness properties make varieties seemingly "manageable" objects, ...
- 2,967
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answers
118
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Representations with small dual
I want to construct an irreducible representation $V$ of any group $G$ such that there exist a mapping of form $\sum_{i=1}^{100} a_i g_i, a_i \in C, g_i \in G$ with kernel of dimension
$dim V-1$. In ...
- 41
1
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157
views
Surjectivity of homomorphism of sheaves
Let $\phi : \mathcal F \to \mathcal G$ be a homomorphism of quasicoherent sheaves on a scheme $X$. What are the necessary properties of the scheme $X$, so that in order to show the surjectivity of $\...
- 11
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votes
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answers
161
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Multiplicity of zero (higher dimensional analog)
Consider a sistem of n holomorphic equations with n unknowns in a neighborhood of zero. Suppose that a solution in a neighborhood of 0 is a k-dimensional manifold.
I want to associate to it some ...
- 61
1
vote
0
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76
views
$H^1$ of a certain line bundle on the resolution of a divisor on a terminal quotient singularity
Let $x \in U$ be a germ of a $3$-dimensional terminal quotient singularity of type $\frac{1}{r} (1,a,r-a)$ over $\mathbb{C}$.
Let $D \in |-K_U|$ be an anticanonical divisor. Assume that $D$ is normal ...
- 909
2
votes
1
answer
161
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Extending kaehler property to desingularizations of quotients
Be $T$ a complex torus (which is not necessarily am abelian variety) of complex dimension $n \geq 2$. On $T$ we have an involution corresponding to the $(-1)$ application (i.e. passing to the inverse ...
- 123
2
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0
answers
112
views
Measuring the obstruction of extending a cover
Let $E$ be a 1-dimensional integral scheme (if you need to assume more, do so). Let $\hat{E}$ be $Spec$ of the completion of a stalk of $E$ at some closed point (think for example $E=Spec(\mathbb{C}[t]...
- 5,929
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145
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unstability of a scroll 2
For the purposes of this question I will work over $\mathbb{C}$. Consider on $T=\mathbb{P}^1$ the bundle $E=O_{T}^{\oplus 3}\oplus O_T(-1)^{\oplus 2}$ and $\mathbb{P}E_T$ the associated projective ...
- 1
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0
answers
79
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Subresultants of primitive polynomials
Given two primitive polynomials $f,g\in D[x]$ over some domain $D$, is there anything we can say about the primitiveness of their $i$-th subresultant polynomials $Sres_i(f,g)$? I.e. is there a simple ...
0
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0
answers
158
views
Are the Dolbeault Operators for a Quotient Space Equivariant?
Let $G$ be a Lie Group and $H$ a closed subgroup such that $G/H$ (the set of right cosets) is a complex manifold manifold. Now $\Omega^1(G/H)$, the space of complex one forms, is a $H$-equivariant ...
- 3,409
4
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158
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Local structure of the surface of bitangents to a quartic
Let $S \subset \mathbb{P}^3$ be a (possibly singular) quartic. I need some information about the local structure of the surface $Bit(S)$ of bitangents to $S$. I have done the computations, but they ...
- 14.7k
1
vote
1
answer
105
views
Is the total space of a module connected?
Let $A$ be a ring and $E$ a module. If $\mathrm{spec} A$ is connected, then so is $\mathrm{spec} S^\bullet E$. If this is not true in general, then what are some minimal conditions that make it true?
- 313
3
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131
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Slicing the fibres of a meromorphic function with the zero set of a section of an ample line bundle
I'm going through a proof of a vanishing theorem by Sommese ($H^{p,q}(X,L) = 0$ for $p+q > n+k$ if $L$ is $k$-ample) and have hit the following brick wall:
I've got a complex projective manifold $...
- 4,343
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0
answers
64
views
Exhaustiveness and regularness of a filtration of a complex
I am learning spectral sequence, but I didn't find a "clear" definition of exhaustiveness and regularness of a filtration $F^{\bullet} := \cdots \subset F^{p+1} \subset F^{p} \subset F^{p-1} \subset \...
- 1,109
1
vote
0
answers
77
views
r-locally linear functions of many variables
Suppose $m,n,r$ are positive integers. Suppose V is a $m$-dimensional vector space over a field F. Let $G(V,n,r)$ denote the space of $n$-tuples of elements of V with the property that the vector ...
- 7,320
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votes
1
answer
107
views
Subgroup Groups and Coordinate Algebra Subalgebras
Let $G$ be a (complex algebraic) group, $H$ a subgroup, and ${\cal O}(G)$ and ${\cal O}(H)$ the coordinate algebras of complex regular functions of $G$ and $H$ respectively. Can ${\cal O}(H)$ ever ...
- 33
2
votes
0
answers
121
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Pole of a function in a diagonal embedding of a normal affine variety.
Let $X$ be a normal affine algebraic variety of dimension $n$ and ${\bar X}^1$, ${\bar X}^2$ be complete normal varieties containing $X$ as a (Zariski) dense open subset. Let $f$ be a regular function ...
- 5,359
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61
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$\mathcal{R}$ is finite over $L_0[e,A]$
Let $\sigma:L \longrightarrow L$ an automorphism of infinite order, where $L_0 \subset L$ is its fixed field. Let $R$ be a commutative subring of $L\{\sigma\}$. Let
\begin{equation}
\mathcal{R}=\...
- 69
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89
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Contraction of extremal ray on a smooth projective threefold
I have some issues about understanding the contraction of extremal ray in a concrete situation:
Let $\mathcal{E}=\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1\times \...
- 41
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0
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44
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Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
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64
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What is the lattice point distribution over binary quadratic forms?
Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.
For simplicity, we keep things only on quadrant I of the ...
