All Questions
22,548 questions
4
votes
1
answer
470
views
To what extent do value sets determine polynomials mod p?
Let $f$ denote a polynomial mod $p$ and associate to it its value set $S_{f} := \{f(x):x \in F_p\}$.
If $|S_{f}|=p$ then $f$ is called a permutation polynomial (because it permutes the elements of $...
- 13k
5
votes
1
answer
213
views
Examples of flat projective morphisms with non-divisorial branch locus
What are some interesting examples of flat projective maps $f : X \to Y$ between smooth varieties such that the discriminant locus $$ \Delta = \{ y \in Y \mid X_y \text{ is singular} \} \subset Y $$
...
- 3,730
2
votes
1
answer
130
views
Where are the critical points of a proper faithfully flat morphism
Suppose that $X$ and $Y$ are compact complex manifolds and $f:X\to Y$ is a faithfully flat map. This map will generally not be a submersion, but it is a submersion away from singular fibres. Assuming ...
- 31
2
votes
0
answers
96
views
On the root numbers of quadruples of quadratic twists of elliptic curves
We got strong numerical evidence for the root numbers and analytic ranks
of quadruples of elliptic curves over the rationals.
Related to this question.
Let $k,k_1,k_2$ be squarefree pairwise coprime ...
- 25.4k
2
votes
1
answer
181
views
Idempotent algebras over absolutely flat ring
Is it possible to classify all idempotent algebras over an absolutely flat ring? Are there any idempotent $E_{\infty}$ algebras which are not discrete?
I am particularly interested in the special case ...
- 2,356
3
votes
1
answer
232
views
Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
- 343
3
votes
0
answers
144
views
Curves on the Hilbert scheme of points on surfaces
Suppose $X$ is a smooth projective surface over $\mathbb{C}$ with irregularity $0$ $(q_1(X)=0)$. I want to understand the curves on the Hilbert scheme of $n$-points on $X$.
By the work of Fogarty, we ...
- 335
7
votes
3
answers
349
views
The rank of elliptic curves and related quadratic twists
Let $E/\mathbb{Q}$ be an elliptic curve, and let $k_1, k_2$ be square-free integers. Can anything be said about the related elliptic curves
$$\displaystyle E/\mathbb{Q}, E^{(k_1)}/\mathbb{Q}, E^{(k_2)}...
- 26.9k
6
votes
2
answers
794
views
Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 673
1
vote
0
answers
60
views
Distribution of the marked points on the components of a stable n-pointed curve of genus zero
Let $\overline{M}_{0,n}$ be the fine moduli space of stable n-pointed curves of genus $g=0$. Let $[(D_{0},p_1,...,p_n)] \in \overline{M}_{0,n}$. Suppose that each component of $D_0$ contains at least ...
- 560
2
votes
0
answers
160
views
Centre of centralisers in connected reductive groups
Let $G$ be a connected reductive group over an algebraically closed field. Let $T$ be a maximal torus and $x\in T$. Let $G_x$ denote the centraliser of $x$ in $G$.
Question: What is an explicit ...
- 2,751
8
votes
0
answers
827
views
Can you glue the Betti $X_\text B$ and de Rham stacks $X_\text{dR}$ together?
Let $X$ be a complex algebraic variety. Is there a (derived prestack) over a base
$$\pi\ :\ X_\text{dR,B}\ \to\ S$$
where $S=\mathbf{R},\mathbf{R}/\mathbf{R}^\times,\mathbf{A}^1_\mathbf{C}/\mathbf{G}...
- 5,711
4
votes
1
answer
215
views
Proper morphism
Maybe this could be a silly question, but I am considering the following problem.
Let $G$ be a connected reductive group and $X$ be an affine $G$-variety, i.e., $G$ acts on $X$. (You can assume the ...
- 147
4
votes
0
answers
125
views
Absolute purity in p-adic case
Suppose that $X$ is a regular proper scheme over $\mathbb{Z}_2$ with smooth generic fiber $X_{\eta}$ and singular special fiber $X_s$. I want to understand $H_{\textrm{et}}^*(X_{\eta},\mathbb{Z}/2)$. ...
- 918
3
votes
0
answers
129
views
Topological interpretation of the existence part of the valuative criterion for properness
Let $X$ be a complex analytic space. I am trying to understand if there is a topological counterpart to the existence part of the valuative criterion for properness. The latter reads: every (ADDED: ...
- 283
5
votes
0
answers
235
views
Triviality of $\unicode{1064}(T_pE ⊗ T_pE)$ for elliptic curves and Bogomolov's lemma
Consider the case of an elliptic curve $E$ over Q, and let $S$ be a finite set of primes including all places of bad reduction and a place $p$ of good reduction.
Bogomolov's Lemma says that when $p$ ...
- 2,907
9
votes
3
answers
584
views
Do rational points on $X(1)$ correspond to elliptic curves up to rational isomorphism?
I'm not sure whether this is obvious or not. The curve $X(1)$ parametrices all elliptic curves up to isomorphism class over $\mathbb{
C}$, and a $K$-rational point corresponds to an elliptic curve ...
- 195
4
votes
1
answer
230
views
Criterion for a rational variety being the projective space
Let $M$ be a smooth projective complex variety. Assume there is an open subscheme $U\subset M$ and an open immersion $U\hookrightarrow \mathbf{P}^n$ such that the codimension of the complements of $U$ ...
- 565
3
votes
1
answer
243
views
Points of multiplicative groups
Let $R$ be a discrete valuation ring with residue field $k$. Denote by $\mathbb G_m:= R[x,1/x]$ the multiplicative group over $R$ and $\mathbb G_{m,k}:= k[x,1/x]$. If $B$ is a flat local $R$-algebra, ...
- 55
1
vote
0
answers
131
views
Can logarithmic blow-ups be constructed étale-locally?
I would like to understand if the logarithmic blow-up of a log scheme can be performed étale locally:
Specifically, suppose that $X^{\dagger}=(X,\mathcal{M}_{X})$ is a fine, saturated log scheme and $...
- 305
3
votes
0
answers
241
views
Generating algebraic points on elliptic curves
Let $E$ be an elliptic curve over $\mathbf{Q}$. One has the modular parameterisation
\begin{align*}
\mathbf{H} \to X_0(N)(\mathbf{C}) \to E(\mathbf{C})
\end{align*}
where $X_0(N)$ is the modular curve ...
- 265
6
votes
1
answer
139
views
Quiver variety, generically symplectic
Theorem 11.3.1 (iv) of "Noncommutative Geometry and Quiver algebras" by Crawley-Boevey, Etingof and Ginzburg claims that, for a dimension vector $\mathbf{d}\in\Sigma_0$, the quiver variety $...
- 985
3
votes
0
answers
227
views
Is it possible to use the Cech complex to compute coherent cohomology in practice?
Suppose I have a closed subvariety $V$ of $\mathbb P^n \times \mathbb A^m$ given by explicit equations. Is it possible in practice to compute the coherent cohomology of $V$ with coefficients in a line ...
- 2,974
13
votes
3
answers
1k
views
$\ell$- vs. $p$-adic and de Rham vs. Betti in the geometric Langlands correspondence
In Emerton–Gee–Hellmann’s IHÉS notes on the categorical $p$-adic local Langlands programme, one finds the following remark:
The differences between the $\ell$-adic and $p$-adic settings are ...
- 607
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 ...
- 5,946
2
votes
0
answers
90
views
What's known about the Steinerian map's indeterminacy locus?
Sorry if this question is asked already, a quick google search didn't yield any answers.
In Classical algebraic geometry, §1.1.6, Dolgachev defines the Steinerian hypersurface $\operatorname{St}(X)$ ...
- 305
2
votes
1
answer
179
views
Picard group of families of smooth projective varieties
Let $f:X \to Y$ be a smooth projective morphism between smooth quasi-projective complex varieties such that fibers of $f$ over closed points are connected.
Assume that $S\subset Y$ is a non-empty ...
- 389
0
votes
0
answers
111
views
Albanese map and curve
Let $S$ be a complex projective integral separated smooth surface (as a scheme). I consider the albanese map $\alpha : S \mapsto A$. I suppose $\alpha(S)$ is a smooth curve of genus $h^{1}(\mathcal{O}...
- 73
2
votes
0
answers
142
views
Computing the coherent cohomology of a quasiprojective variety
I have a quasiprojective variety given by some explicit quations. How do I compute its coherent cohomology (with coefficients in the structure sheaf)? Do I use the Cech complex for an open affine ...
- 2,974
4
votes
2
answers
305
views
Is the property of being a torsor for a smooth affine group scheme detectable on the small étale site?
Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.
Let $h_X$, $h_G$ be the representable sheaves on ...
- 7,537
3
votes
1
answer
483
views
Can you build $\text{Aut}(X)$ using only $\text{QCoh}(X)$?
By the Gabriel-Rosenberg-Brandenberg-Gabber Theorem, sufficiently nice schemes $X$ are determined up to isomorphism by the category $\text{QCoh}(X)$.
My question: for sufficiently nice $X$, can you ...
- 5,711
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
3
votes
0
answers
135
views
The étale fundamental group of an DM stack acting on a locally constant sheaf
Let $W$ be a connected and separated Deligne-Mumford stack of finite type over an algebraically closed field $k$, and $w\in W$ a geometric point. Consider the étale fundamental group $\pi_1^{ét}(W,w)...
1
vote
0
answers
74
views
Does numerical trivial divisor on generic fiber extends to relative numerical trivial divisor?
Let $X, Z$ be complex smooth projective varieties, $f: X\rightarrow Z$ a contraction, and $U\subset Z$ an open subset. Suppose there exists a divisor $L_U$ on $X_U$ such that
$$L_U\equiv _U 0,$$
does ...
- 119
4
votes
1
answer
218
views
Topological interpretation of the canonical cover of a logarithmic Enriques surface
A normal projective surface $Z$ with at worst quotient singularities is called a logarithmic (log) Enriques surface if its canonical Weil divisor $K_Z$ is numerically equivalent to zero, and $H^1(Z,\...
- 213
11
votes
1
answer
463
views
Do finite field point counts remember the singularities of an algebraic variety?
Sorry if this question is terribly naive!
The Weil conjectures famously tell us that if we have a smooth projective variety $X$ defined over the integers with good reduction modulo $p,$ then the Betti ...
- 813
2
votes
0
answers
181
views
Proposition 6.2.3 from Goss "Basic Structures of Function Field Arithmetic"
I'm studying Proposition 6.2.3 from Goss "Basic Structures of Function Field Arithmetic", page 184:
Let $F$ be the sheaf of Data A (a torsion-free coherent $O_{\bar{X}}$-module on $\bar{X}$ ...
- 69
1
vote
0
answers
128
views
Classify all open affine subschemes of a projective variety
Currently I am pondering a question from algebraic geometry that can be stated in very simple terms: Let $X \subseteq P^n_k$ be a projective variety, subscheme of projective $n$-space over an ...
- 708
4
votes
1
answer
136
views
The Poisson center of a symplectic variety
Let $X$ be a symplectic (affine) variety over $\mathbb{C}$, that is, a normal variety with a non-degenerate closed (algebraic) 2-form on the smooth locus. How can we deduce that the Poisson center is ...
- 985
2
votes
0
answers
61
views
Constructing a system of two cubic polynomial equations with exactly 9 real solutions in Maple [closed]
I am trying to construct a system of two cubic polynomial equations in two variables (x and y) with exactly 9 real solutions using Maple. However, I am having trouble finding the appropriate ...
- 21
0
votes
0
answers
176
views
$\mathbb{C}(x,f,g)=\mathbb{C}(x,y)$, with each pair of $\{f,g,x\}$ not generating $\mathbb{C}(x,y)$
Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$.
Write,
$f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$
and
$g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$,
for some $n,m ...
- 2,837
6
votes
2
answers
443
views
Existence of functorial (K-)flat resolutions?
I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe ...
- 91
3
votes
0
answers
82
views
Length of analytic holonomic D-modules
Fix a smooth complex algebraic variety $X$, and $\mathcal D_X$ its sheaf of differential operators.
If a $\mathcal D_X$-module $\mathcal M$ is holonomic, it has finite length. The usual proof uses the ...
- 123
14
votes
0
answers
603
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
- 118k
5
votes
1
answer
176
views
Quotient of the algebra of differential operators by an ideal generated by functions
Say $X$ is a smooth algebraic variety, and write $\mathcal D_X$ for the ring of differential operators on $X$ with the order filtration $F_{\bullet} \mathcal D_X$.
My problem is the following : I have ...
- 123
2
votes
1
answer
304
views
The exact sequence for a derived zero locus
For a locally free sheaf of rank one $L$ on a derived scheme and a morphism $s:L\rightarrow O_{X}$, we consider the derived zero locus of $s$ defined by the following derived fiber product
$$\require{...
- 618
2
votes
1
answer
150
views
Closure of specialization of points of an affine group scheme with smooth generic fiber
Let $R$ be a henselian discrete valuation ring with residue field $k$, and let $G$ be an affine faithfully-flat finite type group scheme over $R$ with smooth generic fiber. Let $R'$ be the ring of ...
- 7,537
3
votes
1
answer
181
views
Probability measure on partition theorem
Probability measure in $\mathbb{R}^1$: Continuous function $f$ that is positive everywhere $\ \int_{-\infty}^{\infty} f dx =1$ (total area under the curve equals 1). For visualization see here.
...
- 53
1
vote
0
answers
248
views
Solving functional analysis problems by using Algebraic geometry
I am thinking about some open problems in nonlinear functional analysis and I just wanted to know if there are any problems that have been solved by using Algebraic geometry techniques in these fields....
2
votes
0
answers
208
views
Fano manifold with vanishing cohomologies of the tangent bundle
Does there exist a projective Fano Manifold $X$ of Picard number one such that $H^i(TX)= 0$ for all $i$, where $TX$ denotes the tangent bundles of $X$ ? If yes, then does there exist any ...
- 549