All Questions
486 questions with no upvoted or accepted answers
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The genus of hyperplane sections
Let $S$ be a connected smooth projective surface over $\mathbb{C}$.
Let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$, and let $d=\dim(\Sigma)$ be the dimension of $\...
- 519
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75
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General fiber and the symmetric product of an ample hypersurface
Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....
- 519
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91
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Closed algebraic subset dominating a curve
In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge
studies in advanced mathematics 76 (2002)" page 228 says:
Let $X$ be a smooth projective variety. If ...
- 519
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276
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Flatness of affine cone due to semicontinuity theorem
I would like to clarify an important aspect from the discussion in this question.
The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic
Geometry Chap. III page ...
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231
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Transversally intersecting divisors $C$ and $D$ in a Hartshorne's AG lemma
Question about proof of lemma V.1.3 in Robin Hartshorne's
Algebraic Geometry on page 358.
Let $X$ be surface. That's for us a nonsingular projective
surface over an algebraically closed field $k$ and ...
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100
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Quotient of $\text{Proj}(A)$ by the action of a finite group
Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$.
I know I can write $X = \bigcup_{...
- 1
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220
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Use of flattening stratification (from Nitsure's construction of Hilbert and Quot schemes)
I study Nitin Nitsures paper on the Construction of Hilbert and Quot Schemes and not understand the propetry (F) completely:
In previous chapter (Embedding Quot into Grassmanian) it was
proved that ...
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89
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Locally freeness of twisted direct images $\pi_* \mathcal{F}(i)$
I have a question about a step in the proof of the
Existence of Flattening Stratification I found in
Nitsure's paper here: https://arxiv.org/abs/math/0504590 This question is closely related to Local ...
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405
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hypersurface of degree d Hilbert polynomial
I am in trouble to solve part (2) Exercise 1.13 from "Moduli of Curves"
by Harris and Morrison on page 9:
Exercise (1.13)
2) Fix a subscheme $X \subset \mathbb{P}^r$. Show, by taking ...
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163
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Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford
In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows.
Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
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221
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Fiber product of singular varieties
Let $f\colon X\to Y$ be a morphism between irreducible $\mathbb{C}$-varieties, such that $f$ is 1-1 and surjective, but $df$ fails to be injective along a subvariety $Z$. (This forces $Y$ to be ...
- 287
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195
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Hyperelliptic Curve (Liu's Book)
Let $Y:=\mathbb{P}^1_k$ and $X$ a hyperelliptic curve. We working with the definition 4.7 from Liu's "Algebraic Geometry and Arithmetic Curves" (page 288)
Namely there exist finite separable map $\pi:...
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153
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Locus of trivialization of an extension of a vector bundle
Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...
- 3,472
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144
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Making a quasi-compact open into an affine open
Let $X$ be a spectral topological space, $U\subset X$ be a quasi-compact open subspace. Is there necessarily some scheme structure on $X$ (we do not require it to be affine) such that $U$ endowed with ...
user138661
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146
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A scheme whose underlying space is the product of the underlying spaces of schemes
We know that the product of two spectral topological spaces is spectral.
If $X$ is the underlying space of the scheme $\mathrm{Spec}\,\mathbb{Z}[x]$, what is a simple example of an affine scheme ...
user140269
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325
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Grothendieck topology on a scheme equivalent to the circle
Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...
user138661
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82
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Bijective restriction of the normalization morphism
Let $X$ be an integral separated scheme of finite type over $\mathbb{C}$. Consider the normalization morphism $f:X'\rightarrow X$. Can we always find an affine open $U\subset X'$ such that $f|_U:U\...
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657
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Mistake in Hartshorne's Exercise II.1.1?
This is really an elementary question, but let me state it. Exercise 1.1 of the second Chapter of Hartshorne's Algebraic Geometry ask to prove that the sheaf associated to the presheaf sending every ...
- 149
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196
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Existence of bijection inducing isomorphism on stalks implies existence of isomorphism
Let $X$, $Y$ be connected smooth projective $\mathbb{C}$-schemes. Let $f:Set(X)\rightarrow Set(Y)$ be a bijection of the underlying sets. Suppose that for any $x\in X$, there exists an isomorphism $O_{...
user138661
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170
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Linear Morphism of Schemes
Let $U$ be an arbitrary scheme and we consider the schematic fiber product $\mathbb{A}^1 _U = \mathbb{A}^1 _{\mathbb{Z}} \times_{\mathbb{Z}} U$.
My question referer to Bosch's "linear morphisms" (of ...
- 5,996
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301
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Irreducible component of a scheme over a dvr
Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
- 155
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188
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constructibility for pushforward
Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...
- 3,472
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183
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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 Spec(Z)....
- 11
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238
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Pro-constructible subset of scheme intersects very dense subsets?
Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?
If $Y$ is just constructible, this is true.
...
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101
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on lifting elements in a tangent space
Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and $...
- 3,472
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67
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open subset in constructible set of divisors
Let a smooth projective curve $X$ over $\mathbb{C}$.
Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$.
Let $N=\deg (D)$ and $X^...
- 3,472
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217
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on the fibers over closed points
Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$
Let $f:X\rightarrow S$ a $k$-morphism of finite type.
We assume that for any closed point $s\in S(\bar{k})$, $...
- 3,472
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259
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How would you call a subscheme of a smooth $S$-scheme?
In my preprint I propose to call $X/S$ quasi-smooth if $X$ can be embedded into a smooth $X'/S$. Does this sound fine?
Upd. So, smoothly embeddable is better? Is it ok to call a morphism smoothly ...
- 16.9k
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103
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Why do I get a morphism $f_P: Spec \mathcal{O}_K \to \mathcal{X}$ for every point $P\in X(K)$? How does this morphism look like?
Hello,
my question probably isn't too hard, but I can't find the answer.
Let $K$ be a number field, $\mathcal{X} /\mathcal{O}_K$ be an arithmetic surface, which is a regular model for a projective, ...
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answers
667
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Quicker way to show that the restriction to a open subvariety is again proper?
Dear all,
Let $f: X \rightarrow Y$ be a morphism of projective varieties over $\mathbb{C}$. Also let $V \subset Y$ be a nontrivial open subvariety and set $U:= f^{-1}(V)$.
I would like to show that $...
- 479
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440
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Foliations in positive characteristic
Ekedahl wrote about foliations in positive characteristic, over the field $\mathbb{Z}/p\mathbb{Z}$ as a subsheaf of the tangent sheaf, that are closed with respect to involution and $p$-power.
My ...
- 1
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0
answers
539
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Projective spaces with nonconstant regular functions
I can construct a scheme by patching that represents a projective space over an arbitrary ring. I can also prove that, if the ring is a Jacobson domain, the only regular functions on it are constants.
...
- 133
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answers
352
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Liftability in positive characteristic
What clsses of algebraic varieties over field of positive characteristic can be lift to $W_2(k)$?
- 85
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524
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DeRham cohomology
The Poincare lemma fails in positive characteristic, since pth powers vanish under differention. My question is : is there still some kind of resolution of the local system k by considering some ...
- 1
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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
-1
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1
answer
895
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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 ...
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