All Questions
1,159 questions
6
votes
1
answer
431
views
Existence of a reduced fiber implies generically reduced (Exercise III-74 from Geometry of Schemes)
This question deals with a concrete exercise from Geomerty of Schemes by Eisenbud and Harris but also moreover the general philosophy attacking typical problems in algebraic geometry of following ...
- 5,986
1
vote
0
answers
210
views
Definition of “morphism of schemes that induces a bijection between irreducible components ”
$\def\sO{\mathcal{O}}\def\sF{\mathcal{F}}$On the Stacks Project there are several instances where the seemingly undefined notion of a “morphism of schemes that induces a bijection between irreducible ...
2
votes
0
answers
194
views
Morphisms $f$ such that $f_* \mathcal O_X$ is a finitely generated $\mathcal O_Y$-algebra
Is there a natural hypothesis that one can put on a finite type morphism $f:X \to Y$ (say $Y$ is locally Noetherian) so that the direct image $f_*\mathcal{O}_X$ is a sheaf of finitely generated $\...
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 412
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 630
1
vote
0
answers
79
views
Is every classical prevariety the set of $k$-rational points of an schematic prevariety? (when $k$ is not algebraically closed)
$\def\cpvar{\mathsf{CPVar}}
\def\spvar{\mathsf{SPVar}}
\def\Spec{\operatorname{Spec}}
\def\class{\mathrm{class}}
\def\sO{\mathcal{O}}
\def\Hom{\operatorname{Hom}}$This question is a follow-up to this ...
1
vote
0
answers
119
views
Action by finite abstract group on affine scheme
Let $X:=\operatorname{Spec}(R)$ an affine Noetherian scheme and $G$ a finite group acting on $X$. Then it is known that the quotient $Y=X/G$ exists as affine scheme $\operatorname{Spec}(R^G)$, let set ...
- 5,986
2
votes
0
answers
147
views
Ramification locus of an integral closure with respect finite field extension
Let $A$ be a Noetherian normal (therefore expecially integral) local ring with unique maximal ideal $\frak{m}$. Let $K$ be it's fraction field, $L$ a finite separable finite field extension of $K$, ...
- 5,986
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
10
votes
1
answer
603
views
Isbell Duality and Dualizing Scheme Objects
I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\...
- 223
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 923
1
vote
0
answers
88
views
Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
1
vote
0
answers
137
views
Pushforward of locally free sheaf by open immersion
Say $X$ is a smooth variety (even just $\mathbb{A}^n$) and $j\colon U\hookrightarrow X$ is an open immersion with $X - U$ of codimension 2 such that $E$ is a locally free sheaf on $U$. Since $X$ is ...
- 43
5
votes
0
answers
220
views
Is there a simple counterexample to étale proper base change on the unbounded derived category?
The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
- 605
2
votes
1
answer
94
views
Base change for fundamental group prime to p in mixed characteristic?
I found the answer to this question while typing it up, but since I've already written it, it is probably worthwhile to post-and-answer in case someone finds it useful.
Let $S=\operatorname{Spec}\...
- 371
2
votes
2
answers
406
views
Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?
Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ ...
- 413
3
votes
1
answer
517
views
Classical $k$-prevarieties vs reduced $k$-schemes of finite type. What happens when $k$ is not algebraically closed?
$\def\cpvar{\mathsf{CPVar}}
\def\spvar{\mathsf{SPVar}}
\def\Spec{\operatorname{Spec}}
\def\class{\mathrm{class}}
\def\sO{\mathcal{O}}
\def\Hom{\operatorname{Hom}}$Let $k$ be a field. By classical $k$-...
2
votes
0
answers
107
views
Deformation of complex manifolds that admit reduction modulo $p$
Let $(M,B,\omega)$ be a complex analytic family of compact (projective non singular) complex manifolds, where $B \subset \mathbb{C}^{m}$ is some domain. Lets consider a subclass of such manifolds $\{...
- 331
2
votes
0
answers
85
views
Bialynicki-Birula decomposition for $\mathbb{G}_m$-actions on projective schemes
The classical BB-decomposition works for non-singular projective varieties. Here I want to consider projective schemes, in particular when the scheme is not reduced.
Let $\Bbbk=\mathbb{C}$. Let $X$ be ...
- 930
1
vote
0
answers
150
views
Relative compactification without resolutions of singularities
Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
- 309
3
votes
0
answers
112
views
What are the possibilities of the general fibres in an Iitaka fibration?
This question is motivated by complex algebraic geometry.
If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
- 9,501
4
votes
0
answers
184
views
smooth super scheme which is not smooth
I am following the very nice "Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes" by Bruzzo, Ruiperez and Polishchuk. I am having some problem in order to give ...
- 558
4
votes
0
answers
103
views
Boundary divisor of an equivariant compactification of a non-trivial twist of $\mathbb{G}_a^n$
Let $K$ be a non-perfect field of positive characteristic. Then there exist non-trivial forms of $\mathbb{G}_a^n$ which split over finite purely inseparable extensions of $K$, i.e., algebraic groups $...
3
votes
0
answers
94
views
Dimension of a kernel of a cocycle map
Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following:
Compute the kernel (or at ...
- 911
3
votes
0
answers
122
views
Torsion of Fermat hypersurfaces
An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group,
$$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$
where $K = k(X)$ is the function ...
- 3,730
6
votes
0
answers
205
views
Quadric contain tangent variety of a curve in $\mathbb{P}^5$
Let $Q^4 \subset \mathbb{P}^5$ a smooth quadric over $\mathbb{C}$
which is via Pluecker map isomorphic
to Grassmannian of lines $\mathbb{G}(1,\mathbb{P}^3)$
in $\mathbb{P}^3$.
Consider following ...
- 619
3
votes
0
answers
87
views
Obstruction for points to be contained in smooth hypersurfaces in tterms of inseparability degree of residue field
Let $k$ be an imperfect field of char $p>0$ and
$x \in \mathbb{P}^n_k$ be closed point of projective space.
In this discussion Qing Liu wrote that
Over an imperfect field, a reduced point can not ...
- 619
1
vote
1
answer
226
views
flatness of restriction of structure sheaf over ring of global sections
Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...
2
votes
0
answers
119
views
Regular hypersurface containing a point of a variety $X$ over perfect field $k$
Let $X$ be a variety over perfect field $k$ and $x \in X$ some closed reduced
point. (at this point I'm not 100% percent sure if it's neccessary to assume $x$ to be reduced, ie that it's stalk is ...
- 619
3
votes
0
answers
105
views
Points with residue fields having big inseparability degree cannot be contained in smooth hypersurfaces
Let $X$ be a $k$-scheme over imperfect field $k$ and $x \in X $ some (reduced) point with residue field $\kappa(x) = \mathcal{O}_{X,x}/ \mathfrak{m}_x$. How to check that if $\kappa(x)$ has "big ...
- 619
1
vote
0
answers
29
views
Connected components of Isotropy types as strata of Poisson leaves
Let $X$ be a smooth affine variety with an algebraic symplectic form $\omega$. Let $G$ be a finite subgroup of the group of symplectomorphisms of $X$.
We can say that $X$ is trivially a normal variety ...
- 609
2
votes
0
answers
167
views
How to compute the $G$-theory of this simplicial toric surface?
Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma_0$ be the cone in $\mathbb{R}^2$ generated by $e_1,e_2$.And let $\sigma_1$ be the cone in $\mathbb{R}^2$ generated by $e_2,-...
- 639
0
votes
1
answer
146
views
Are projective bundles corresponding to non-isomorphic vector bundles always non-isomorphic?
Suppose we are given a scheme $S$ and two vector bundles $V$ and $W$ over $S$. Is it always true that $\mathbb{P}(V)\cong \mathbb{P}(W)$ implies that $V\cong W$ as $S$-schemes?
If the statement is ...
- 187
2
votes
2
answers
320
views
How to compute the $G$-theory of the variety $\mathbb{P}^1\times\mathbb{P}^1$?
Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be the fiber product of two copies of $\mathbb{P}^1_k$ over the affine scheme $\operatorname{Spec}(k)$.I am trying to compute ...
- 639
1
vote
0
answers
120
views
Linear span of tangential variety
Let $X \subset \mathbb{P}^N$ be a projective variety of dimension $n$. Let us denote with $TX=\bigcup_{x \in X}\mathbb{T}_xX$ the tangential variety, where $\mathbb{T}_x X$ is the projective tangent ...
- 1,343
1
vote
1
answer
203
views
Reduction step to $k=\bar{k}$ in the proof of rigidity lemma
I do not understand the following proof in the paper Abelian varieties by Edixhoven, van der Geert, and Moonen:
(1.12) Rigidity Lemma. Let $X$, $Y$ and $Z$ be algebraic varieties over a field $k$. ...
- 5,986
0
votes
1
answer
270
views
Proof of rigidity lemma
I have problems to understand a proof in this paper by Pierrick Dartois on Abelian varieties:
Theorem 1.13 (rigidity lemma). Let $ \varphi: X \times_k Y \to Z$ be a morphism of $k$-schemes. Assume ...
- 5,986
2
votes
0
answers
119
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
2
votes
0
answers
99
views
Geometric generic point of a complete linear system
In the following context: Let $S$ be a connected smooth projective surface over $\mathbb{C}$, and let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$. Let $d=\dim(\Sigma)$ ...
- 519
0
votes
1
answer
384
views
Relation between canonical bundles under étale maps
Let $X$ and $Y$ be two integral separated Noetherian Gorenstein schemes over a base field $k$ of arbitrary characteristic whose local rings are unique factorization domains and $f: X\to Y$ an étale ...
- 5,986
8
votes
1
answer
342
views
The Grothendieck topology of closed immersions on schemes
Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$...
- 329
1
vote
1
answer
694
views
Under what conditions is an open subscheme of an affine scheme affine and what ring corresponds to it?
It is well known that an open subscheme of an affine scheme is not necessarily an affine one. But what are (if possible the most general) sufficient conditions for its affinity? And is it known how, ...
- 6,962
6
votes
0
answers
191
views
Computing the automorphism scheme of projective space
$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...
- 605
2
votes
1
answer
273
views
Most general lifting property for proper morphisms
Let $\mathcal C$ be the class of morphisms $f\colon U\to V$ of schemes such that for every proper map $g\colon X\to Y$ between schemes and every commutative solid square
there exists a lift $h$ ...
4
votes
0
answers
77
views
Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 12.9k
4
votes
1
answer
360
views
Construct morphisms of schemes on level of associated functors
I have a general question about techniques used in @Emerton's proof, sketched below, in the answer to $\mathbb{P}^n$ is simply connected.
Given a finite étale map $\pi: Y \to \mathbb P^n$ (we regard ...
- 5,986
0
votes
0
answers
267
views
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
4
votes
2
answers
642
views
Basic question on projective bundles
Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
- 663
1
vote
1
answer
88
views
Generic finite subgroups, associated to small finite fields, of reductive algebraic groups
Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says:
Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
- 12.9k
2
votes
0
answers
292
views
Cartier and the continuity of the early history of schemes
If you allow me I would divide the early history of schemes this way
_ Weil, Zariski, Bourbaki, Nagata, Van der Waerden,... up to Chevalley (you can find an interesting blog here)
J P Serre varieties ...
- 894