All Questions
8,187 questions with no upvoted or accepted answers
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167
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Canonical bundle and diagonal morphism on varieties
Hi,
this question concerns the canonical bundle $\omega_X$ on a smooth projective $k-$variety $X$.
I want to consider the diagonal embedding
$i:X\rightarrow X\times X$
on $X$.
Furthermore denote ...
- 183
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0
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214
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How do I find non-linear sets that are invariant under a certain linear transformation?
I have an invertible linear transformation $T:F^k\to F^k$, where $F$ is a finite field and $k$ is a natural number.
It's easy to find the linear subspaces S that are invariant under T.
How do I find ...
- 179
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0
answers
616
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About automorphisms of ratonal surfaces.
Hi. I have a question about automorphisms of smooth ratonal surfaces. (I am not an algebraic geometer, so my question could be stupid. I am sorry about that.)
Let $X_k$ be a blow-up of $\mathbb{P}^2$ ...
- 1,037
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252
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local hyper tor is zero
Hallo,
in a proof I read an argument which used the following "fact" which was no further explained:
if you have a closed point $x$ on a smooth projective variety over a field $X$ and denote with $k(...
- 623
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146
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$\mathbb{P}^1 \times \mathbb{P}^1$ bundles and genus 2 curve fibrations
I've been constructing genus 2 curve fibrations by starting with a (3,2) curve in $\mathbb{P}^1 \times \mathbb{P^1}$ and then promoting its scalar coefficients to sections of line bundles over some ...
- 175
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166
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Can the zero-degree part of $M_f \otimes_{S_f} N_f$ be identified with $M_{(f)} \otimes_{S_{(f)}} N_{(f)}$?
The isomorphism ${(M \otimes _ {S} N)} _ {f} = M _ {f} \otimes _ {S _ {f}} N _ {f}$ is well-known. Here, $S$ is a graded ring, and $M,N$ are graded $S$ modules.
Now, let $f$ be any homogeneous ...
- 945
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330
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Algebraic independence of two polynomials in $\mathbb{C}[x, y]$ and Combinatorial Nullstellensatz.
Let $f(x, y), g(x, y) \in \mathbb{C}[x, y]$. Consider the image $S$ of the map
$(f, g): \mathbb{C}^2 \rightarrow \mathbb{C}^2, (z_1, z_2) \mapsto (f(z_1, z_2), g(z_1, z_2)).$
If for an integer $d \...
- 704
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votes
0
answers
309
views
Weight filtration of MHSs
This is probably a very stupid question, but could someone explain to me where the weight filtration of mixed Hodge structures come from and why we actually need it?
If the Hodge-to-de Rham spectral ...
- 191
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0
answers
237
views
resolution of singular points on plane curves and base change
Let $k$ be a field and $C/k$ be an affine plane curve over $k$, namely $C = \mathrm{Spec}(A)$ for some $A = k[x,y]/(f(x,y))$, here $f(x,y) \in k[x,y]$ is an irreducible polynomial. Let $B$ be the ...
- 1,109
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261
views
Is an immersed Kronecker join always a multilinear variety on a Hilbert space?
The question asked is:
Is the implicitization of an arbitrary-rank immersed Kronecker join always a multilinear variety on a Hilbert space?
This is related to another MathOverflow question
In ...
- 1,389
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votes
0
answers
352
views
Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
0
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0
answers
221
views
Representable morphisms of Artin stacks.
Given a representable morphisms $f:\mathcal{X}\to\mathcal{Y}$ of Artin stacks, do the fibers of $f$ have always nonnegative dimension? If not, can you give me some examples of what can happen?
- 773
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votes
0
answers
283
views
Singular fibers of Kodaira dimension
I will be pleased if someone could give me some good reference to understand Singular fibers of Kodaira type. In fact I_6, I_3 and A_1. Thanks
- 645
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0
answers
272
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A simple(possibly trivial) question about Grothendieck Topologies
Let $C$ be the category of sets. Define coverings {$U_i\to U$} to be jointly surjective maps, i.e. $U$ is the union of the images of $U_i$. Then if $F$ is a sheaf of sets on $C$, is it clear that $F(\...
- 1,563
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198
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why a reduced ring can be embedded into a sum of integral rings?
Hi,
the question is exactly
"why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?"
Is this simply because in the normalization process we can have many irreducible ...
- 141
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answers
425
views
help needed with a proof in <etale cohomology> by milne
P23: Let $f:Y\rightarrow X$ be locally of finite-type. Prove that if $\Delta:Y\to Y\times Y$ is an open immersion, then $f$ is unramified.
$\newcommand{\spec}{\operatorname{spec}}$
... we reduce the ...
- 21
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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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197
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Existence of flat models of a smooth finite type algebra over $R((t))$
Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary),
$B$ an algebra of finite type over ring of the formal Laurent series $R((t))$, which is smooth.
Up to this generality, can one ...
- 51
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answers
1k
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fibre of a morphism
Let $f : X\rightarrow Y$ be a morphism of scheme. For any point $y\in Y$, the fibre of $f$
over $y$ is defined to be $X_y = X\times_Y Spec(k(y))$. Then the underlying set of $X_y$ is
bijective with $f^...
- 135
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answers
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
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273
views
Ext groups and flat modules on a K3 surface
Let $S$ be a $K3$ surface and $X$ the moduli space of some stable sheaves on it. Let $G$ be the universal family on $X\times S$ and $F$ the ideal of section of $X\times S\to X$. Knowing that for every ...
- 773
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answers
297
views
higher direct image: proof of the proper case.
Hi.
Let $f:X→S$ be a proper, open, surjective morphism of complex reduced spaces with constant fiber dimension n (or universally open morphism with n-fibers between locally noetherian excellent ...
- 435
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0
answers
333
views
Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff
Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic ...
- 1,499
0
votes
0
answers
830
views
Finding components of a preimage
Let $f:X\to Y$ be a degree $d$ morphism of complex projective varieties, and let $V\subset Y$ an irreducible subvariety, $W$ its preimage under $f$. I want to find all of the components of $W$.
...
- 16k
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votes
0
answers
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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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
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votes
1
answer
126
views
Integer quadratic representation subject to discriminant minimization algorithm
Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers.
More concretely, is there an algorithm to find $...
0
votes
1
answer
568
views
Non-cohomological proof that a noetherian scheme $X$ is affine if its reduction $X_{red}$ is affine
Can we prove that a noetherian scheme $X$ is affine if its reduction $X_{red}$ is affine without using cohomology?
Remark
Here is a similar question.
- 1,169
-1
votes
1
answer
555
views
Noetherianity assumptions in Hartshorne's book
It seems that noetherian assumptions are not necessary in many results by Hartshorne, in his book "Algebraic Geometry". How much is this true? Could you please give examples?
- 1,422
-1
votes
1
answer
895
views
Restriction of a Cartier divisor
Let $X$ be a surface (so $2$-dimensional proper $k$-scheme)
$D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and
$C \subset X$ a closed ...
- 6,048
-1
votes
1
answer
99
views
Finitely many subvarieties as divisor
Let $X$ be a smooth projective variety over an algebraically closed field of characteristic $0$ and of dimension $n\geq 10$. Let $(C_i)_{1\leq i\leq N}$ (resp. $(S_i)_{1\leq i\leq N}$) be smooth ...
- 1,463
-1
votes
1
answer
504
views
Dominant morphism of Affine Scheme
Suppose to have $\phi$ a ring morphsim from $A$ to $B$, let $X=SpecA$ , $Y=SpecB$ and $\psi$ the induced morphism of affine schemes. It's true that if $\psi$ dominant than $\phi$ is injective?
- 29
-1
votes
1
answer
333
views
Quantum cohomology of isomorphic Poisson varieties
This question is related with my previous one Quantum cohomology rings as invariants, but now, I want to ask a more concrete thing. If $X$ and $Y$ are Poisson varieties which are isomorphic (as a ...
- 1
-1
votes
1
answer
537
views
Relative flatness
Can someone me say if this (perhaps obvious!) claim is true:
let $f:X\rightarrow S$ be an open, surjective morphism of complex spaces reduced or without embedded components and with $n$-pure ...
- 435
-1
votes
1
answer
265
views
About sufficient condition for smoothness
Dear Brian.
The dimension 1 case is very special. We assume $X$ no compact (Stein if we want), normal and of dimension >1...
In fact, i want to prove the following:
Let $f:X\rightarrow S$ be an ...
- 435
-1
votes
1
answer
259
views
Pure Quotient and pure sub-object
Let $\mathcal{C}$ be the category of modules over a ring.
Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ ...
- 3
-5
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0
answers
126
views
Is a quiver variety a moduli stack of quiver representations?
As the title, I was just wondering is a quiver variety a moduli stack of quiver representations? I am familiar with quiver varieties but not that familiar with moduli stacks, so I was just wondering ...
- 133