All Questions
8,187 questions with no upvoted or accepted answers
14
votes
0
answers
958
views
What is the state of art in Groebner bases
How big polynomial systems can we deal with? How do you know when you don't even have to try?
Motivation:
Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...
- 8,597
14
votes
0
answers
914
views
The symmetric group and the field with one element
I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of ...
- 20.2k
14
votes
0
answers
2k
views
conformal blocks for beginners
I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...
- 3,779
14
votes
0
answers
638
views
Are stable limits of complex curves acquiring a singularity obtained by capping off the Milnor fiber and attaching that to the normalization?
Background on Milnor fibers:
Suppose a smooth complex plane curve acquires an isolated singularity f(x,y) = 0 with Milnor number µ = dim.C[[x,y]]/(∂f/∂x, ∂f/∂y), and r local branches. Then we know ...
- 12.4k
14
votes
0
answers
547
views
When are the fibers of a resolution of singularities reduced?
I apologize if this is too much of a fishing expedition, but I've had bad luck searching for any literature on this subject, and I was hoping someone could tell me if it's too easy to be worth ...
- 44.7k
14
votes
0
answers
3k
views
Ample divisors on projective surfaces
Question: If $X$ is a projective surface and $U$ is an open affine subset of $X$, then is it true that $X \setminus U$ is the support of an (effective) ample divisor on $X$?
Background: I was reading ...
- 5,359
14
votes
0
answers
899
views
Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
13
votes
0
answers
261
views
Big list of Hochster dual concepts
Let $X$ be a spectral space. Then there is a canonical space $X^\vee$ with the same points, same constructible topology, and the opposite specialization order. This is known as “Hochster duality”, and ...
13
votes
0
answers
451
views
Have the details in Voevodsky's original approach to Milnor's conjecture been filled in?
In Bloch-Kato conjecture for $\mathbb{Z}/2$-coefficients and algebraic Morava $K$-theories Voevodsky claimed a proof of Milnor's conjecture under some assumptions. Later he came up with a different ...
- 441
13
votes
0
answers
995
views
What is a BPS state and why is it the cohomology of a moduli space?
The notion of a BPS state has existed in physics for a long time: I do not understand it completely, but I get the impression they are the supersymmetric analogue of ground states, and then physicists ...
- 5,711
13
votes
0
answers
786
views
Seek for a algebro-geometric proof: the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective
It is a well-known fact that the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective.
What I want is a proof by method of algebraic geometry. ...
- 1,168
13
votes
0
answers
290
views
The inequality $K_X^n\leq (-n-1)^n \chi (\mathscr{O}_X)$
Let $X$ be a smooth projective manifold of dimension $n$ with $K_X$ ample. Does the inequality
$$K_X^n\leq (-n-1)^n \chi (\mathscr{O}_X)$$ hold? For $n=2$ or 3 it is
equivalent to the Miyaoka-Yau ...
- 38k
13
votes
0
answers
663
views
On a kind of Hilbert irreducibility theorem
Let us work over a number field $k$. Let $C$ be a non-empty open subscheme of $\mathbb{P}^{1}_{k}$, and $X\to C$ a family of smooth, projective hyperbolic curves such that $X(k)\to C(k)$ is surjective....
- 1,000
13
votes
0
answers
319
views
Exotic smooth structures on Fano manifolds
If two Fano projective manifolds are homeomorphic are they diffeomorphic?
There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of ...
user164740
13
votes
0
answers
749
views
Rings whose Frobenius is flat
Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.
I am interested in rings for which $F_R$ is flat (hence ...
anon1432
13
votes
0
answers
321
views
Varieties isomorphic $\mathrm{mod}\:p$ are diffeomorphic
If two smooth proper varieties over $\mathbb{Q}$ have isomorphic smooth reductions modulo some prime (for some choice of integral models) are they diffeomorphic after tensoring with $\mathbb{C}$?
user145520
13
votes
0
answers
524
views
Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?
Consider two functors given by $R \mapsto GL(R((h)))$ and $R \mapsto PGL(R((h)))$ for a ring $R$. It is easy to see that these functors are sheaves in Zariski topology (in fact for any affine variety $...
13
votes
0
answers
498
views
Do regular functions separate points?
Suppose $K$ is a field, $V$ is a $K$-variety (finite type reduced separated $K$-scheme), and $p,q$ are $K$-points of $V$. Must there be an open subset $U$ of $V$ containing both $p$ and $q$ and a ...
- 1,689
13
votes
0
answers
414
views
The Barr-Boole-Galois topos; a modification of sets to play well with schemes
William Lawvere, in his 2015 CT talk "Alexander Grothendieck & the Concept of Space", introduced the "Barr-Boole-Galois topos", which plays the role that sets do for topological spaces (with ...
user30211
13
votes
0
answers
503
views
Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
- 6,622
13
votes
0
answers
834
views
Where is a full proof of the precise Torelli theorem in the literature?
The precise form of Torelli's theorem is as follows (translated from Serre's appendix to Lauter - Geometric methods for improving the upper bounds on the number of rational points on algebraic curves ...
- 3,539
13
votes
0
answers
348
views
An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?
I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian.
Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...
- 3,539
13
votes
0
answers
260
views
Alexander modules and weight filtrations
$\def\Ext{\mathrm{Ext}}$I have the following situation: I have a complex algebraic variety $X$. I want to compute the weight filtration on various abelian covers $Y \to X$. As a concrete example, take ...
- 156k
13
votes
0
answers
564
views
Cohomology of a blow-up of a real algebraic variety
Let $X$ be a complex algebraic variety, $Z \subset X$ a closed subvariety, $\mathrm{Bl}_Z X$ the blow-up and $E$ the exceptional divisor. There is an isomorphism of cohomology groups
$$ H^k(X(\mathbf ...
- 40.2k
13
votes
0
answers
743
views
Kähler-Ricci flow approach for Beauville-Bogomolov type decomposition?
Is there any Kähler Ricci flow method for solving structure theorems in Algebraic geometry
In fact If $X$ be a Calabi-Yau manifold then we can descend the Kähler Ricci flow to its finite etale ...
user21574
13
votes
0
answers
606
views
Algebraic closure of a field in constructive mathematics
There is a short note by André Joyal called Les théorèmes de Chevalley-Tarski et remarque sur l'algèbre constructive (pp. 256-258). It is claimed there that there is a constructive version of the ...
- 4,459
13
votes
0
answers
596
views
What does deformation theory have to do with Serre duality?
The cotangent bundle $\Omega_X$ of a smooth scheme $X$ shows up in two places in my understanding of algebraic geometry. The first is deformation theory, where maps out of $\Omega_X$ control the ...
- 8,004
13
votes
0
answers
740
views
Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.
$D^b_{...
- 9,019
13
votes
0
answers
2k
views
Calculation-free proof of the Weyl Integral formula for U(n)
The Weyl integral formula states that if $f$ is a class function on $U(n),$ $T$ is the torus of diagonal matrices in $U(n)$, and $dU(n)$ and $dT$ are the standard Haar measures on $U(n)$ and $T,$ then:...
- 28.1k
13
votes
0
answers
1k
views
Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{n-1}$
Let $f,g:\mathbb{C}^n\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^n\to \mathbb{C}^n$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and $...
- 131
13
votes
0
answers
408
views
Does the de Rham version of Cohen's theorem hold in the $\infty$-setting?
One of the first results that one needs to prove in the theory of chiral algebras is a de Rham version of Cohen's theorem on the homology of $C_n$ spaces. This is achieved in Beilinson-Drinfeld's book ...
- 3,242
13
votes
0
answers
504
views
Other examples of the algebro-geometric Ran space
First off, sorry if this seems vague.
Let's recall some definition. Let $X$ be a curve over a field $k$ and $G$ an algebraic group, then the space $Ran_G(X)$ as defined by Lurie in his Tamagawa ...
- 2,327
13
votes
0
answers
282
views
Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles
Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A ...
- 5,504
13
votes
0
answers
411
views
Topological type of Brieskorn manifolds
Let us consider the complex hypersurface and suppose that $n\geq 3$:
$$F(d,n)=\{(z_0,\ldots,z_n)\in \mathbb{C}^{n+1}:z_0^d+z_1^d+\ldots+z_n^d=0\}$$
and the link $V(d,n)=F(d,n)\cap S^{2n+1}_{\epsilon}$ ...
- 9,870
13
votes
0
answers
892
views
Stack of Tannakian categories? Galois descent?
I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\...
- 13.3k
13
votes
0
answers
474
views
Refinement of concept of support of a module
My rings are commutative and noetherian.
The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This ...
- 8,727
13
votes
0
answers
375
views
Complex manifold which is algebraic away from codimension \ge 2
If $X$ is a complex manifold and $Z$ is a closed subset of codimension $\ge 2$ such that $X-Z$ has an algebraic structure, then is $X$ algebraic; i.e. a scheme? If not is $X$ an algebraic space?
...
- 3,968
13
votes
0
answers
595
views
What's about N. M. Katz' "over-world" of exp. sums?
Having just read in N. M. Katz' beautiful old survey on exponential sums a d differential eq.s, I wonder what became out of his question (on p. 297 - 300) on a "general conceptual framework in which ...
- 10.8k
13
votes
0
answers
2k
views
Applications of cohomology and base change?
What is the theorem on coherent cohomology and base change good for?
One version of the theorem is:
Suppse $f \colon X \to Y$ is a proper morphism of noetherian schemes and $F$ is a $Y$-flat coherent ...
13
votes
0
answers
714
views
Non-unital algebraic geometry
In his great answer to this MO question James Borger gives a geometric characterization of non-unital $k$-algebras (i.e. not necessarily unital) : They correspond to affine schemes $X$ over $k$ ...
- 63.1k
13
votes
0
answers
849
views
Is every Abelian variety a direct summand of the Jacobian of a curve?
Though the answer is "yes" for Abelian varieties up to isogeny, the stronger statement is, perhaps, false. Does anyone know a counterexample?
- 131
13
votes
0
answers
943
views
Beilinson-Bernstein localization in positive characteristic
This is a follow-up to this question; in particular, I'm wondering if anyone can expand upon the interesting answers given by Kevin McGerty and David Ben-Zvi there. (In particular, in this question I'...
- 3,637
12
votes
0
answers
605
views
What's a holonomic D-module from the point of view of de Rham spaces?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. We can consider its de Rham space $X_\text{dR}$ as the sheaf on $(\textsf{Sch}/\mathbb{C})_\text{ét}$ defined by $X_\text{dR}(S):= X(S_\text{...
- 771
12
votes
0
answers
2k
views
Roadmap to geometric Langlands for a mathematical physics student
I am a student of both mathematics and physics, who has recently been studying string theory. My mathematics background is mostly differential geometry (principal bundles, Lie groups, etc.), although ...
- 349
12
votes
0
answers
284
views
Regular two-dimensional algebraic spaces
Let $X$ be an algebraic space which is integral, noetherian, separated, two-dimensional and regular. We keep these assumptions throughout.
Question 1. Is $X$ always a scheme?
Question 2. If $X$ is a ...
- 19.6k
12
votes
0
answers
247
views
Symmetric spaces are quandles. Is this important?
For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
- 2,516
12
votes
0
answers
435
views
History of use of "=" symbol to mean "is canonically isomorphic to"
Let $A$ be a commutative ring, and let $f$ and $g$ denote elements of $A$ such that the prime ideals of $A$ containing $f$ are precisely the prime ideals containing $g$ (a not completely trivial ...
- 41.4k
12
votes
0
answers
222
views
Spin 6-fold with signature $\pm 16$
Does there exist a spin (i.e. $\frac{c_{1}}{2} \in H^{2}(M,\mathbb{Z})$) smooth complex projective $6$-fold with signature $\pm 16$?
The motivation is the Rochlin-Ochanine theorem, which says that $16$...
- 6,995
12
votes
0
answers
388
views
Perverse sheaves and representation theory
At my university we are having a working group on perverse sheaves, with the aim of applying them to representation theory (Lusztig canonical bases for quivers/quantum groups etc). We are still ...
- 1,061
12
votes
0
answers
410
views
Formal versus algebraic good reduction of abelian varieties
Let $k$ be a non-archimedean complete field with ring of integers $k^\circ$, and let $A$ be an abelian variety over $k$. Let $\mathscr A$ be a formal abelian model of $A$; i.e., $\mathscr A$ is an ...
- 1,584