All Questions
1,159 questions
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414
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Is the following morphism etale
Let $Y$ be a reduced noetherian $1$-dimensional scheme such that the normalization morphism $f:X \longrightarrow Y$ is finite. Let $g:Y\longrightarrow Z$ be a finite flat morphism, where $Z$ is a ...
- 3
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0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
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votes
0
answers
112
views
Irregularity of surfaces for dominant maps
I have a question about an argument in the proof of Lemma 1.2.(1) in Quotients of K3 surfaces modulo involutions by D. Q. Zhang:
Let Let $(X, \sigma)$ be X be a smooth projective K3 surface with an ...
- 6,006
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votes
0
answers
99
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Quotients of K3 surfaces vs cyclic covers
Let $X$ be an algebraic K3 surface (for sake of simplicity, with base field of char $\neq 2$) and $f: X \to X$ a non-symplectic morphism (i.e. non-symplectic in sense of that that the induced action ...
- 6,006
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111
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Lefschetz Theorem in Dolgachev's On automorphisms of Enriques Surfaces
Let $F$ be a Enriques surface over $\Bbb C$. I have a question about a detail in the proof of Proposition 2.1. from Dolgachev's On automorphisms of Enriques surfaces.
This 2.1. Proposition. states ...
- 6,006
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127
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Relative minimal models of pencils of surfaces
I would like to ask for recomendation for literature on theory relative minimal models of surfaces, where "relative" in sense of that the study objects are not surfaces alone (="absolue ...
- 6,006
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124
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Counit map surjective
Let $X \to Y$ a (set theoretically) surjective morphism of schemes, $L$ a line bundle/invertible sheaf on $X$ (maybe more generally a locally free coh sheaf, but let's stick firstly on invertible ...
- 6,006
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118
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Induced action on infinitesimal thickenings by an algebraic group
Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
- 6,006
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109
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Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
- 91
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331
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Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 149
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145
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Stalk at isolated point of proper map with $f_*\mathcal{O}_X=\mathcal{O}_Y$
Let $f:X \to Y$ a proper surjective map between schemes $X,Y$ with additional assumption $f_*\mathcal{O}_X=\mathcal{O}_Y$. Let $y \in Y$ with a 'isolated point fiber', i.e., $f^{-1}(y)=\{x\}$ as set. ...
- 6,006
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votes
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answers
267
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completion and tensor product
Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue ...
- 3,472
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votes
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190
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How to compute the exceptional divisor of this blow-up
Suppose that $k$ is a field and $R$ is the ring $k[x,xy,xy^2,xy^3]$.Let $I$ be the maximal ideal of $R$ generated by $x,xy,xy^2,xy^3$.Let $E$ be the exceptional divisor of the blow-up of Spec$R$ along ...
- 639
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130
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Is a closed subsecheme contained in a Cartier divisor?
Let $X$ be a variety over a field $k$. For a closed subscheme $Z\hookrightarrow X$ and a closed point $x\in X$ such that $\text{codim}_XZ \geq 1$ and $x\notin |Z|$, is there an effective Cartier ...
- 349
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224
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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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votes
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answers
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
views
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 ...
- 6,006
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votes
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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 ...
- 6,006
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votes
0
answers
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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answers
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 ...
- 6,006
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votes
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answers
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 ...
- 6,006
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votes
0
answers
405
views
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 ...
- 6,006
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163
views
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 ...
- 3,625
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answers
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:...
- 6,006
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answers
153
views
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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votes
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answers
144
views
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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votes
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answers
325
views
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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votes
0
answers
82
views
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
views
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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0
answers
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
0
votes
0
answers
170
views
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 ...
- 6,006
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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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votes
0
answers
188
views
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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votes
0
answers
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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votes
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67
views
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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votes
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217
views
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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answers
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, ...
- 1
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votes
0
answers
668
views
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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0
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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
0
votes
0
answers
539
views
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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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
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0
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524
views
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
0
votes
0
answers
151
views
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
votes
1
answer
191
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Number of distinct scheme structures on a set [closed]
Given a cardinal number $|X|$, how many isomorphism classes of schemes with the cardinality of the set of points equal to $|X|$ are there?
user142033