All Questions
1,159 questions
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
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
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
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
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 ...
- 39.3k
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 $k$. ...
- 12.9k
0
votes
1
answer
271
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 ...
- 6,006
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 ...
- 28.8k
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, ...
- 28.8k
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 $S$...
- 1,329
2
votes
3
answers
1k
views
Weil restriction
I've already asked a similar question in SE, without success, so I've decided to post here a more general version of my question.
Let $f: Y \to X$ be a finite étale morphism of smooth proper varieties ...
- 12.9k
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 ...
- 396
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 $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
- 28.8k
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
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
- 63.9k
3
votes
0
answers
127
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,541
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
- 63.9k
0
votes
0
answers
190
views
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
1
vote
0
answers
88
views
Invariance of numerical class of a curve in Higgs-Grassmann schemes
Premise
Let $X$ be a projective variety of dimension $n\geq1$ over an algebraically closed field of characteristic $0$.
A Higgs sheaf $\mathfrak{E}$ is a pair $(E,\varphi)$ where $E$ is a $\mathcal{...
- 828
4
votes
0
answers
64
views
An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
1
vote
0
answers
44
views
What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?
$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...
- 605
6
votes
0
answers
130
views
Epimorphisms and quotients in Sch versus $\mathrm{Sh}(\mathrm{Ring}^{\mathrm{op}},\mathrm{Zar})$
$\DeclareMathOperator\Ring{Ring}\DeclareMathOperator\Aff{Aff}\DeclareMathOperator\op{op}\DeclareMathOperator\Sch{Sch}\DeclareMathOperator\Zar{Zar}$The category of schemes sits, fully faithfully, in ...
- 63.9k
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
5
votes
1
answer
827
views
Can the functor of the points of a scheme be characterized by its values on subcategories of the affine schemes?
A scheme is equivalent to a functor $\mathcal{F}:\textbf{AffSchemes} \rightarrow \textbf{Set}$ such that it admits a cover of affine schemes and is a sheaf of rings on the Zariski site.
Suppose $\...
- 1,270
1
vote
1
answer
149
views
When is $R$ a direct summand of Frobenius pushforwards?
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
- 20.5k
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
0
votes
0
answers
130
views
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
1
vote
0
answers
106
views
Joins of (closed) subschemes and Zariski-local Z-functors
$\newcommand\Aff{\mathrm{Aff}}\newcommand\cRing{\mathrm{cRing}}\newcommand\Sch{\mathrm{Sch}}$Equip $\Aff = \cRing^\text{op}$ with the Zariski Grothendieck-topology. There are nested categories:
$$\Aff\...
- 12.9k
4
votes
1
answer
227
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
1
vote
0
answers
205
views
Projective scheme over the integers
Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?
- 653
2
votes
0
answers
221
views
Is this blow-up a line bundle over the projective line
Let $R$ be the ring $\mathbb{C}[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $I$ be the ideal of $R$ generated by $a,d$. Let $X=\operatorname{Spec}(R)$ and $\tilde{X}$ be the blow-up of the affine scheme $X$ ...
9
votes
0
answers
1k
views
Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 12.9k
6
votes
1
answer
771
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 149k
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 2,821
4
votes
1
answer
219
views
Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?
Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$.
...
- 4,492
1
vote
1
answer
163
views
Arithmetic ampleness and scalings of the metric
Let $\overline L= (L, h)$ be a hermitian $C^
\infty$ line bundle on an arithmetic variety $X\to\operatorname{Spec }\mathbb Z$ (I am reasoning in terms of higher Arakelov geometry, like in Gillet & ...
- 408
1
vote
0
answers
114
views
Check whether a closed point of a Noetherian affine scheme is a local complete intersection
Suppose that $k$ is an algebraically closed field and $A$ is the ring $k[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $X$ be Spec$A$, and $m$ be the maximal ideal of $A$ generated by the quotient images of $a,...
- 639
2
votes
0
answers
253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 63.9k
2
votes
1
answer
178
views
Constructibility of the locus of points where the fiber is an isomorphism modulo nilpotents
Let $f: X \rightarrow S$ be a finitely presented morphism of schemes and let $$E = \{s \in S \mid \text{ $X_s$ is a point with residue field $\kappa(s)$ } \}$$
Is $E$ a constructible set?
The basic ...
- 19.6k
1
vote
0
answers
122
views
How to compute the G-theory groups of a blow-up of Noetherian schemes
Suppose that $k$ is an algebraically closed field and $R$ is a finitely generated $k$-algebra such that if $X$ denotes Spec$R$, then the only closed, singular point of $X$ is the origin. Let $\tilde{X}...
- 639
4
votes
1
answer
620
views
Normal schemes and Serre's criterion
Serre's criterion says that for a scheme to be normal is equivalent to it being $R_1$ (i.e. regular in codimension $1$) and $S_2$ (i.e. regular functions on $X-Y$ extend to $Y$ if $Y$ has codimension ...
- 39.3k
0
votes
0
answers
224
views
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