All Questions
22,546 questions
6
votes
1
answer
293
views
When does isomorphism on singular cohomology imply isomorphism on Picard and Brauer groups?
Assume that $f:X\to Y$ is a morphism of complex varieties, and the homomorphisms $H^i_\text{sing}(f)$ are bijective for $0\le i\le 3$ (though possibly $3$ is too much here:)). Under which ...
- 16.9k
1
vote
0
answers
71
views
Computing the symmetric product of a sheaf supported on a divisor
Let $Y\hookrightarrow X$ be a inclusion of a smooth divisor $Y$ (i.e. codimension 1 closed subscheme) in to smooth variety $X$. Let $\mathcal{L}$ be a vector bundle over $X$, we regard it as a locally ...
- 653
4
votes
0
answers
214
views
Algebraic logic in the style of algebraic geometry
I am writing a thesis on algebraic logic, I wonder if there is any recent research on an idea mentioned in Yuri Manin's book on algebraic geometry and in another Russian textbook on differential ...
- 558
24
votes
1
answer
868
views
The congruence subgroup property for mapping class groups and a conjecture of Grothendieck
This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.
Recall that a subgroup $K$ of a ...
- 25k
3
votes
1
answer
231
views
Are principal parabolic group scheme bundles Zariski locally trivial?
Let $P$ denote a parabolic subgroup scheme of $\operatorname{Sp}(2n;F)$, where $F$ is a field (I am interested in $K=\mathbb{Q}_p$ so possibly okay to assume local with characteristic $0$ if it makes ...
- 2,907
2
votes
1
answer
233
views
existence of a coherent sheaf
I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\...
- 613
3
votes
1
answer
292
views
Derived Koszul complex
Let $X$ be a projective variety over $\mathbb{C}$ and $V$ be a vector bundle over $X$. Let $\pi: V\to X$ be the natural projection.
Let $i: X\to V$ be the zero section map. Let $V^\vee$ be the dual ...
- 653
2
votes
0
answers
129
views
Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?
$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
- 9,019
2
votes
0
answers
109
views
Punctured neighbourhood of quotient singularity is not simply connected?
Let $X$ be a variety (irreducible, normal) over a field $k$ which is algebraically closed with characteristic $0$. Suppose that $X$ has only one singular point $p\in X$, so $Y:=X\setminus p$ is smooth....
- 131
5
votes
1
answer
261
views
Principal bundles over smooth projective curve
Let $X$ be a smooth and connected projective curve over $\mathbb{C}$ and $G$ a reductive connected group over $\mathbb{C}$. Fix a faithful representation $G \subseteq \mathrm{GL}_n$.
Given a $G$-...
- 1,061
4
votes
1
answer
165
views
Describing the compactified Jacobian of a nodal curve
$\DeclareMathOperator{\Pic}{Pic}$Let $C$ be an integral projective curve over $\mathbb C$, which is smooth except for a single node $x\in C$. Let $M$ be the moduli space of stable torsion-free sheaves ...
- 1,286
4
votes
0
answers
97
views
Is there a concept of a map of Grothendieck sites having dense image?
Someone recently asked if one can talk about a map being etale dense just like one can talk about it being Zariski dense. My main question is: has anyone discussed such a notion?
On a simple ...
- 15.4k
2
votes
0
answers
123
views
Isomorphism between motivic cohomology and algebraic cobordism
Let $MGL$ be the algebraic cobordism defined by Voevodsky, and $\Omega$ the algebraic cobordism constructed by Levine and Morel. For motivic cohomology $H^{p,q}$, we use Suslin-Voevodsky's definition. ...
- 121
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
3
votes
0
answers
156
views
A possible application of deformation theory?
Let $f : \mathbb{R}^{n} \to \mathbb{R}$ be a real-valued polynomial function. Consider the family of real algebraic sets:
$$
V_c = f^{-1}(c), \quad c \in (-1,1).
$$
I am interested in determining how ...
- 357
2
votes
1
answer
401
views
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 2,964
3
votes
0
answers
150
views
$p$-adic points of open subschemes of complete intersections
I'm currently studying the $3\times 3$ magic square of squares problem for a research project. The variety is initially given by the intersection of $8$ quadrics in $\mathbb{P}^8$, but via Gröbner ...
- 31
4
votes
1
answer
328
views
Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 41
4
votes
1
answer
418
views
Definition of Chow quotient
I am reading M. M. Kapranov's paper "Chow quotients of Grassmannians. I." (English) in Sergej Gelfand (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in ...
- 41
4
votes
1
answer
243
views
On the degeneration of the elliptic surface $E(n)$
The following matter should be widely known (if true). I am sorry for my ignorance!
For the natural $n$, let $E(n)$ be the corresponding elliptic surface.
In the analytic world, there exists a well-...
- 153
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
0
votes
1
answer
93
views
Maximal number of couples of vectors $(r,n)$ such that the product $\langle n_i , n_j\rangle+\langle r_i \times n_i , r_j \times n_j\rangle<0$?
Let $\langle v,w\rangle$ and $v\times w$ stand for the dot product and the cross product of vectors $v,w\in\mathbb R^3$.
Do there exist $2k$ vectors
$$
r_1, \dots, r_k, \; n_1, \dots, n_k\in \mathbb{...
0
votes
0
answers
104
views
Non-degenerate bilinear pairing of finite dimensional algebras
A finite dimensional algebra (over $\mathbb{C}$, say) is said to be Frobenius if it comes equipped with a nondegenerate bilinear pairing
\begin{align*}
\langle -, - \rangle : A \times A \rightarrow \...
- 71
5
votes
1
answer
237
views
Methods of finding integer solutions beyond the reach of direct search
Consider a classical problem: given a polynomial Diophantine equation $P(x_1,\dots,x_n)=0$, determine whether it has an integer solution. While this problem is undecidable in general, we may still ...
- 7,182
11
votes
6
answers
2k
views
Hard problems with an easy-to-understand answer
I am very interested by problem in mathematics which are difficult (go at least 10 years without a resolution, say) but which have a solution that is short and elementary.
In this video Launay gave an ...
4
votes
1
answer
297
views
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
Let $X$ be a normal algebraic surface (over $\mathbb{C}$) and $Y$ its smooth locus, i.e., the complement of the singularities of $X$. Suppose $Z\subset Y$ is a Zariski open subset of $X$. Then is it ...
- 335
2
votes
1
answer
200
views
Ampleness of the pullback of the relative dualizing sheaf of $\overline{\mathcal{M}_{g,n+1}}\rightarrow\overline{\mathcal{M}_{g,n}}$
There is a natural map $f : \overline{\mathcal{M}_{g,n+1}}\rightarrow\overline{\mathcal{M}_{g,n}}$ identifying the source with the universal family over the target. Let $\sigma_1,\ldots,\sigma_n$ be ...
- 10.7k
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
1
vote
0
answers
73
views
Is the subgroup of $R$-trivial classes of an algebraic group an algebraic subgroup?
Let $G$ be an algebraic group over a field $F$. I'm willing to assume it is linear if that changes anything to what I'm going to say, and even reductive if that helps (but I don't think it should make ...
- 272
0
votes
1
answer
302
views
Is a bijective regular map between affine varieties a homeomorphism?
Let $X \subset \mathbb{A}^n,~ Y \subset \mathbb{A}^m$ be affine varieties. Consider a regular map $f: X \to Y$. If $f$ is bijective, can we conclude that $f$ is an open mapping w.r.t the Zariski ...
1
vote
0
answers
97
views
Weil restriction of cycles and norm algebra
This question is on a concrete descrption of weil restricton of an affine algebra.
Let L/K be a Galois extension. Since I only care about the quadratic case, we may assume that $\Gamma:=\operatorname{...
2
votes
1
answer
154
views
$R^1\Gamma = 0$, and the Mumford stability
Let $S$ be a smooth projective surface with an ample divisor $H$ so that $K_S \cdot H < 0$.
Let us consider the Mumford slope $\mu(E) = \frac{H \cdot c_1(E)}{\operatorname{rk}(E)}$ on $\...
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
1
vote
1
answer
140
views
Specialization of points on the generic fiber in a prestable model of $\mathbb{P}^1$
Let $R$ be a complete DVR with uniformizer $\pi$, fraction field $K$ and residue field $k$. We assume $k$ is algebraically closed.
Let $X$ be a prestable model of $\mathbb{P}^1_K$ over $R$, so $X$ is ...
- 7,527
3
votes
0
answers
195
views
Are local complete intersections of small codimension necessarily (global) complete intersections?
Hartshorne's 1974 conjecture states that a smooth closed subvariety $X$ of $\mathbb{CP}^r$ of dimension $>2/3r$ is necessarily (globally) a complete intersection. Is anything interesting known ...
- 16.9k
2
votes
0
answers
144
views
Picard group of the category of numerical motives
Is anything known about the Picard group of $Chow_{Num}(k, \mathbb{F}_{p})$ (numerical Chow motives with $\mathbb{F}_{p}-$coefficients)?
Perhaps the Picard groups of some other categories of pure ...
- 87
2
votes
0
answers
153
views
Uniqueness and existence of maps
I am currently reading the Berkeley lectures on Perfectoid Spaces by Scholze and Weinstein. In the section "The adic open unit disk over $\mathbb{Z}_p$" we encounter from Proposition 4.2.6 ...
0
votes
1
answer
119
views
Detecting singular points from a parametrization
Suppose $r(t)$ parametrizes some, say algebraic, curve in the plane. It can certainly be that $r$ is smooth but the curve is not, since $r$ resolves double points by passing through them at different ...
- 13
5
votes
0
answers
217
views
Lifting a morphism between quasi-projective varieties
Let $\mathcal{V}$ be an affine algebraic variety over $\mathbb{R}$, $G$ be a finite group acting freely on $\mathcal{V}$. Consider the quotient space $Y:=\mathcal{V}/G$, which itself is a quasi-...
- 364
10
votes
2
answers
286
views
Finding a model for $X_G$ with $G\subseteq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$
I'm studying modular curves as part of my doctorate work and would like to understand how one gets from a subgroup $G$ of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$ to an equation for $X_G$. By $...
- 195
11
votes
0
answers
183
views
Are algebras with rational structures dense in varieties of real Lie nilpotent algebras?
One says that a real nilpotent Lie algebra has $\mathbb Q$-structure if it has a basis with rational structure constants. It is well known that there are nilpotent Lie algebras without $\mathbb Q$-...
- 723
1
vote
1
answer
238
views
Generalization of Gromov-Witten theory counting surfaces, 3-folds, etc
I don't work on the Gromov-Witten theory, but I find that I need to study the following problem, which seems to be similar to the Gromov-Witten theory:
Let $X$ be a variety and $\alpha_{1}, \cdots, \...
- 305
7
votes
0
answers
249
views
Phantoms and Geometry
Let $\mathcal{D}(X)$ be the bounded derived category of coherent sheaves on a smooth projective variety $X$. An autoequivalence $\Phi: \mathcal{D}(X) \to \mathcal{D}(X)$ is called phantom if it ...
- 71
8
votes
0
answers
402
views
Langlands program in higher dimensions
We can view the Langlands program in each of its versions (local/global, arithmetic/geometric) as giving a description of the finite-dimensional representations of the étale fundamental group of a ...
- 91
2
votes
1
answer
328
views
Extension by zero operation
Suppose you have a closed subset $Z$ of a topological space $X$, and $F$ is a sheaf on $Z$. Then one can consider the extension by zero sheaf $F^X$ on $X$.
What are some examples and situations which ...
- 129
3
votes
0
answers
389
views
Status of motives in higher category theory: motives and algebraic cycles through a higher categorical perspective
A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives.
Primary question:
Have there been any recent developments/advances on the above question? If not,...
2
votes
1
answer
330
views
Completion of a local ring is noetherian (under some hypothesis)
I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly:
Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
- 293
5
votes
1
answer
364
views
Unramified fppf cohomology
Let $F$ be a number field, $\mathcal{O}_F$ its ring of integers and $G$ an fppf $\mathcal{O}_F$-group scheme.
See the question Unramified Galois cohomology of number fields for unramified cohomology ...
- 265
6
votes
1
answer
307
views
Hochschild cohomology and differential operators
The Hochschild-Kostant-Rosenberg theorem says, that for a commutative algebra $R$ over a field $k$ with certain smoothness and finiteness, we have an identification $\mathrm{HH}^\bullet(R)\cong \...
- 985