All Questions
1,159 questions
2
votes
0
answers
91
views
Deductions from the pushforward of the structure sheaf being the structure sheaf
Note: I originally asked this on MSE without any success.
Let $f\colon X\to Y$ be a morphism of schemes for which $f_*(\mathcal{O}_X)=\mathcal{O_Y}$. When this condition arises has been discussed in ...
- 411
3
votes
0
answers
73
views
p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
7
votes
0
answers
295
views
+100
Zariski connectedness theorem: from analytic & topological viewpoint
Let $p:Y \to X$ be a proper, flat (see later why) surjective map between smooth connected complex varieties $Y,X$, esp. $X$ unibranch. Assume that there exist a Zariski open (esp. dense) $U \subset X$ ...
- 6,016
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 6,016
5
votes
4
answers
1k
views
Stable points in GIT: geometric picture
Is there a geometric picture justifying why "stable points" in GIT (Geometric Invariant Theory) are actually called "stable"? Stable, with respect to which effect? (Here, I have ...
- 619
14
votes
1
answer
569
views
What is the "schematic" point of view for regular polyhedra?
Last week, I read Wikipedia's article on Alexander Grothendieck. It lists his twelve greatest contributions to mathematics as accounted for in Grothendieck's own Récoltes et Semailles. The final item ...
1
vote
0
answers
57
views
Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$.
In "Singularity of Minimal Model Program" by Janos kollar p249, it says,
If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
- 328
2
votes
1
answer
271
views
Jacobian fibration of elliptic fibration: basic relations between Enriques invariants
Let $f: X \to B$ be an elliptic fibration, so proper map from smooth surface $X$ onto smooth conn. curve over alg closed base field $k$ with connected fibers such that almost all fibers are elliptic ...
- 6,016
1
vote
0
answers
109
views
L.c.i locus of Hilbert scheme of points on singular varieties
Let $X$ be an algebraic variety over $\mathbb{C}$. What can we say about the l.c.i. locus of $\text{Hilb}^n(X)$?
When $X$ is smooth, it is well-known that the l.c.i. locus of $\text{Hilb}^n(X)$ is ...
- 385
1
vote
0
answers
70
views
Descent of $G$-invariant formal system of parameters using GAGF
Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
- 6,016
3
votes
1
answer
191
views
Irreducibility under etale ring map
Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$.
If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
- 328
1
vote
0
answers
219
views
Quotient of K3 surfaces by non-symplectic automorphism of finite order
Let $X$ be a $K3$ surface and $f: X \to X$ a non-symplectic morphism (ie non symplectic in sense of that that the induced action on $H(X,K_X=H^0(X, \Omega_X^2)$ is not trivial) of finite order.
...
- 6,016
0
votes
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
0
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 ...
- 6,016
2
votes
0
answers
105
views
Torsion Freeness of Sheaf of Kähler Differentials
Let $X$ be an irreducible scheme over some base field $k$. Consider the sheaf of Kähler differentials $\Omega_{X/k}$. Let $w: \Omega_{X/k} \to j_* \Omega_{K(X)/k}$ be natural map induced by enbedding ...
- 6,016
1
vote
0
answers
161
views
Special elliptic pencil of an Enriques surface (arguments in a proof)
I have a couple of questions about arguments in the proof of Lemma 2.6 (see absol page 199, rel p 9) from Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
The setup: Let $Y$ ...
- 6,016
0
votes
0
answers
111
views
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 ...
- 6,016
1
vote
0
answers
117
views
Quotient of K3 surface: complex vs positive characteristic
Let $f: X \to X$ be a non-symplectic automorphism of finite order of complex projective K3 surface $X$. (Recall: Non-symplectic means that the induced action on $H(X,K_X)=H^0(X, \Omega_X^2)$ is not ...
- 6,016
2
votes
0
answers
92
views
Lifting smooth proper varieties over finite fields to finite extensions of $W(k)[1/p]$
Let $k$ be a finite field of characteristic $p > 0$, and let $X$ be a smooth proper variety over $k$. It is generally unknown whether $X$ admits a smooth proper lifting over $W(k$, where $W(k)$ ...
- 519
0
votes
0
answers
127
views
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 ...
- 6,016
5
votes
1
answer
568
views
Dualizing sheaf of nodal curve
Let $C$ be a connected, nodal (I'm working with definition from Alper's notes on Stacks & Moduli, see p 210), projective curve over an alg closed field $k$, beeing everywhere smooth except at a ...
- 6,016
0
votes
0
answers
99
views
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 ...
- 6,016
0
votes
0
answers
124
views
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 ...
- 6,016
1
vote
0
answers
85
views
Relation between quot scheme of birational curve
I am very new to algebraic geometry. Currently reading about Hilbert and quot scheme. I want to know more about the structure and properties of Hilbert and quot schemes over curves. My question is the ...
- 629
4
votes
1
answer
252
views
Multiplicative cancellation for trivial vector bundles
Let $X$ be a scheme, ${\mathscr L}$ an invertible ${\mathscr O}_X$-module, and $d$ a positive integer. If ${\mathscr L}^{\oplus d} \simeq {\mathscr O}_X^{\oplus d}$, does it follow that ${\mathscr L} \...
- 318
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
4
votes
1
answer
293
views
Counter example to every closed subscheme $\operatorname{Proj} A$ is of the form $\operatorname{Proj}A/I$
I was under the impression that for a positively graded ring $A$ (not necessarily generated in degree $1$) that every closed subscheme of $\operatorname{Proj}A$ was of the $\operatorname{Proj}A/I$. ...
- 391
2
votes
0
answers
154
views
A schematic representability of an algebraic space with group action
In the book "Néron Models" (BLR), there is a statement as follows (on page 164):
Let $S$ be a locally noetherian scheme and let $G$ be a smooth algebraic group space over $S$ with connected ...
- 291
4
votes
1
answer
236
views
Ampleness verifiable over faithfully flat cover
Let $X$ be a Noetherian scheme over a field $k$ and $\mathcal{L}$ an invertible sheaf. Recall $\mathcal{L}$ is called ample iff for every coherent $\mathcal{M}$ there exist a $n_0(M)$ such that for ...
- 6,016
1
vote
1
answer
211
views
Characterize descents of geometric finite étale cover by means of homotopy exact sequence
Let $X/k$ be a geometrically connected $k$-variety (=separated of finite type, esp quasi-compact; the base field $k$ assumed to be separable, so $\overline{k}=k^{\text{sep}}$), $\overline{X} := X \...
- 6,016
1
vote
0
answers
178
views
Prop 1.3 in "Birational geometry of algebraic varieties": specialization of rational curves
I have a couple of questions about some arguments in proof of Proposition 1.3 from Birational geometry of algebraic varieties by Kollár and Mori (p 8):
Proposition 1.3. [Abh56, Prop. 4] Let $X$ be ...
- 6,016
4
votes
0
answers
178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 6,016
4
votes
1
answer
446
views
Exact functor in syntomic cohomology
By Tag 04C4 of the Stacks Project, for $f:X\rightarrow Y$ a closed immersion of schemes, the pushforward $f_*$ is exact for abelian sheaves on the big syntomic site.
Is it also true for a finite flat ...
- 3,472
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
1
vote
1
answer
591
views
Argument for non-existence of elliptic curve over $\mathbb{C}[t, t^{-1}]$
I have a couple of questions about this answer by Noam D. Elkies showing that there exist no elliptic curve $E$ over $\mathbb{C}[t, t^{-1}]$ having nonconstant $j_E$-invariant.
The strategy is to ...
- 6,016
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
5
votes
1
answer
351
views
Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"
I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...
- 6,016
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
182
votes
33
answers
32k
views
What should be learned in a first serious schemes course?
I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons (...
2
votes
1
answer
143
views
$G$- Fixed Point Scheme explicitly
Let $G$ be an abstract finite group acting on a separated $k$-scheme $X$. ($k$ a field; note we can canonically promote $G$ to a $k$- scheme). Then a result by Demazure and Grothendieck (in "...
- 6,016
2
votes
0
answers
171
views
Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
- 6,016
3
votes
0
answers
171
views
Nice blowups are universal algebraic fiber spaces?
We say that a proper (maybe projective) morphism $f:X \to Y$ is a universal algebraic fiber space if $f_* O_X = O_Y$ holds universally. (This means: for any morphism $Y' \to Y$, if $X' = Y' \times_Y X$...
- 39
5
votes
1
answer
288
views
Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
- 6,016
1
vote
1
answer
110
views
Torsor of finite presentation and surjectivity of map of $\overline{k}$-valued points
I have a question about the content of remark 2.6.6. (i) (p 18) from M. Brion's notes on structure of algebraic groups.
Let $G$ be a group scheme over certain fixed base field $k$ (as all other ...
- 6,016
1
vote
2
answers
197
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
7
votes
0
answers
277
views
Is every normalization a blowup?
I asked this at math.stackexchange, but received no reply.
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
...
1
vote
0
answers
104
views
Calculation of intersection multiplicity after the restricting to a fiber
Let $X\to\operatorname{Spec} \mathbb Z$ be an arithmetic surface which is projective, regular and integral. Let $D$ and $E$ two divisors intersecting at a point $x\in X$ that lies over the prime $p$. ...
- 321
1
vote
1
answer
367
views
Proj construction and nilpotent homogenous elements in graded ring
Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...
- 6,016
6
votes
1
answer
269
views
Criteria for when Gauss-Manin sheaves are vector bundles
Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
1
vote
0
answers
71
views
Explicit field extension for semistable models of curves
The paper arxiv:1211.4624 briefly summarizes the way to find a semistable model of a curve $X/K$ (the existence of the model is ensured by the Deligne-Mumford theorem). Specifically the author says ...
- 321