All Questions
624 questions
3
votes
0
answers
42
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
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=\...
- 5,986
0
votes
0
answers
89
views
Extend algebraic morphism to a compactification with normal crossing boundary
Suppose $X$ and $Y$ are smooth algebraic variety over a char $0$ field $k$, and $f:X\to Y$ a morphism. I want to ask whether there exists compactifications $\bar X$ and $\bar Y$ such that $\bar X\...
- 785
1
vote
0
answers
92
views
Compactification of smooth varieties with normal crossing boundary
I see in this paper, page 46, the second sentence of 4.1, that every smooth variety over a characteristic $0$ field can be embedded into a proper smooth variety with normal crossing boundary, and the ...
- 785
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
0
votes
0
answers
152
views
Compactification of the Jacobian of singular curves via parabolic modules
I would like to better understand a certain compactification of the Jacobian variety of a singular algebraic plane curve as described in Cook's Ph.D. 1993 thesis Local and Global Aspects
of the Module ...
- 1
1
vote
0
answers
88
views
Equivariant resolution of singularity making a pullback of a line bundle admit a root
I am considering the following situation. Let $X$ be an irreducible normal projective variety with an action of a linear algebraic group $H$, and we have a $H$-equivariant line bundle $L$ over $X$. We ...
- 323
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
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
2
votes
1
answer
117
views
Blow up of terminal singularity and canonical singularity
A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if
$(i)$
it it is $\mathbb{Q}$-Gorenstein. and
$(ii)$For any resolution of singularity $F:Y\rightarrow X$,
$K_Y-f^*K_X>...
- 328
6
votes
1
answer
487
views
Variety without a compactification whose complement is smooth
Let $X$ be a smooth, separated complex algebraic variety. By Hironaka, there exists a compactification $j : X \to \bar{X}$ of $X$ so that $\bar{X} \setminus X$ is a simple normal crossings divisor.
Is ...
- 813
1
vote
1
answer
92
views
Explicit expression of simultaneous resolution of semi-universal deformation of ADE singularity
Denote the semi-universal deformation of ADE singularity by $\mathcal{Y}\to\mathfrak{h}^{\mathbb{C}}/W$, where $\mathfrak{h}^{\mathbb{C}}$ is the complex Cartan algebra of root system of type ADE and $...
- 93
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
0
answers
62
views
About the definition of cDV singularity
M. Reid defines cDV singularity as follow in his paper "CANONICAL 3-FOLDS"
A point $p\in X$ of a 3-fold is called a compound Du Val point if for some section H throgh $P$, $P\in H$ is a Du ...
- 328
2
votes
0
answers
108
views
Finiteness of rational double point
Let $(R,\mathfrak{m
})$ be a three dimensional complete local ring over a field $k$ of arbitrary characteristic and let $f\in R$ and $R/f$ is a rational double point.
My question is as follows.
Are ...
- 328
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 $k^\times$
$$
\mathbb{...
- 53
3
votes
1
answer
197
views
Terminal singularities of fibers vs total space
Suppose $f\colon X \to Y$ is a flat map of complex varieties (or more generally DM stacks?). Suppose every fiber has at most terminal singularities and that $Y$ is smooth. Under what conditions is it ...
- 43
1
vote
1
answer
151
views
Examples of small resolutions in dimension 4 and higher
I have seen numerous examples of 3-folds admitting small resolutions. Are there similar examples in higher dimension? In particular, I am looking for examples of singular varieties of dimension $4$ ...
- 2,323
3
votes
1
answer
246
views
Borel-Moore homology for resolution of singularities
Let $X$ be a singular projective variety. Denote by $Z$ the singular locus of $X$. Consider the resolution of singularities $$\pi: \widetilde{X} \to X$$
Denote by $E$ the exceptional divisor. We know ...
- 2,323
2
votes
1
answer
176
views
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
- 185
3
votes
1
answer
197
views
Image, upto direct summands, of derived push-forward of resolution of singularities
Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
- 480
4
votes
2
answers
435
views
Question about surface singularities
Throughout, $X$ will be a projective surface. I am looking for examples of the following surface singularities,
I) A rational singularity that is not quotient. Obviously, it has to be non-Gorenstein, ...
- 345
2
votes
0
answers
113
views
Singularities of curves over DVRs with non-reduced special fibre
Let $R$ be a complete DVR of mixed characteristic with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p>0$. Suppose that $\mathcal{X}$ is a normal $R$-curve such ...
- 305
4
votes
0
answers
108
views
Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
2
votes
1
answer
106
views
Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$
Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I am looking for a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
- 2,521
4
votes
0
answers
135
views
Nilpotent orbits in characteristic $0$ vs. positive characteristics
Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
- 2,751
2
votes
1
answer
214
views
Perfect complexes of plane nodal cubic curve
Let $C\subset\mathbb{P}^2$ be a plane nodal cubic curve with a unique singular point $O$ at the origin. Then I consider its normalization, denoted by $\widetilde{C}$ and let $\pi:\widetilde{C}\...
- 1,982
1
vote
0
answers
61
views
Do local and global symplectic resolutions have same monodromy?
Yoshinori Namikawa associates a Weyl group $ W $ to any symplectic affine complex variety $ X $ with good $ \mathbb{C}^* $-action. He provides a semi-explicit description of $ W $, which requires ...
4
votes
0
answers
183
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 41
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
3
votes
1
answer
163
views
Can Coulomb branches have symplectic resolutions?
My question is about Coulomb branches of a $3D$ $\mathcal{N}=4$ supersymmetric gauge theory, in the sense of Bravermann, Finkelberg and Nakajima Towards a mathematical definition of Coulomb
branches ...
- 3,318
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
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
1
vote
0
answers
128
views
Log resolution of ideal and associated dual graph
Let $(X,0)$ be a complex surface germ with an isolated singularity and $I$ be an $\mathfrak{m}\text{-primary}$ ideal (contains a power of the maximal ideal $\mathfrak{m}$) of the local ring $\...
- 25
1
vote
0
answers
48
views
Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?
Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 64k
1
vote
0
answers
66
views
How can I calculate the derivative of an integral with respect to a parameter if Leibniz's formula gives a divergent integral?
We are working on the problem related to a magnetic field in an axially symmetric magnetic plasma trap. Let's express the vector potential through the magnetic flux function
\begin{gather}
\label{1:01}...
- 241
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 ...
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
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
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
3
votes
0
answers
119
views
Semi-stable model over a totally ramified extension
Notation: Let $R$ be a DVR, $K=\text{Frac}(R)$ and $k=R/\mathfrak{m}$. Given an $R$-scheme $X$, write $X_K=X\times_{R} K$ for the generic fiber and $X_k=X\times_R k$ for the special fiber.
Suppose $k$ ...
- 282
2
votes
1
answer
327
views
Krull dimension of the smooth locus
Let $R$ be a normal complete local domain of dimension $n \geq 4$. Does there exist a prime ideal $\mathfrak{p}$ of height $\dim(R) - 1$ such that $R_{\mathfrak{p}}$ is a regular local ring? In ...
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
2
votes
1
answer
200
views
Extending étale covers from the regular locus to a resolution of singularities
Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of ...
- 3,730
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
2
votes
1
answer
331
views
Lindelöf paper on meromorphic singularities
Does anyone know a digital link to the following paper, written by Ernst Lindelöf:
"Mémoire sur certaines inégalités dans la théorie des fonctions monogènes,
et sur quelques propriétés nouvelles ...
- 232
2
votes
1
answer
304
views
Normal forms of ADE singularities
Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar ...
- 481