All Questions
8,187 questions with no upvoted or accepted answers
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Rational points of weighted projective spaces
[EDIT (Feb. 27, 2024): No answer to the reference question yet, but I explain a more general statement at the end.]
Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive ...
- 19.6k
12
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0
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347
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Quivers as noncommutative curves
I've heard that an idea behind noncommutative geometry (in dim 1) is to study "noncommutative" analogues of $\text{Coh}(\text{curve})$, rather than the curve directly. Apparently the ...
- 5,711
12
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0
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586
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Global version of Gabber's rigidity theorem
I had a question regarding Gabber's rigidity.
Let $A$ be a ring (let's assume Noetherian) and $I$ be an ideal, since the pair $(\hat{A},I)$ is a henselian pair ($\hat{A}$ is the completion along $I$), ...
- 5,901
12
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676
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Kihara-like Z/6Z elliptic curve families
Shoichi Kihara constructed a family of elliptic curves with Mordell–Weil group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}^3$ (generic rank at least 3) in 2006. Kihara's family produces a number of rank 8 ...
- 913
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553
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When does Matiyasevich's theorem "kick in"?
Hilbert's 10th problem was famously resolved by the Matiyasevich–Robinson–Davis–Putnam theorem: the theorem implies that there is no algorithm which decides whether a given polynomial equation with ...
- 26.9k
12
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288
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birational geometry of moduli spaces: why work on the coarse space?
In studying the birational geometry of $\overline{\mathcal{M}}_g$, it seems standard to work with the coarse space $\overline{M}_g$ rather than the smooth stack $\overline{\mathcal{M}}_g$. Why is this?...
- 735
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689
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Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'
It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version ...
- 439
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325
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Is there an algorithm to compute a Belyi map for the Riemann surface?
Let $y^2=x^5-x-1$ be an affine model of a projective complex curve, is there an algorithm to compute the Belyi map (preferably of small degree), i.e., map to the projective line ramified only at $\{0,...
- 269
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2k
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Visualization and new geometry in higher stacks
I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond ...
- 787
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265
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Mixed characteristic analogue of algebraicity of the diagonal of two-variable power series?
Let $f=\sum_{n,m \geq 0}^{\infty}[a_{nm}]p^ny^m \in \mathbb Z_p[[y]]$, where $a_{nm} \in \mathbb F_p$ and $[\cdot]$ means the Teichmüller lifting. Define $I(f)=\sum_{n \geq 0}[a_{nn}]p^nt^n \in \...
- 6,622
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346
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Is there an odd degree unirational parametrization of a cubic threefold?
A cubic threefold is a smooth degree $3$ hypersurface in $\mathbb{P}^4$. Is there a cubic threefold $X$ over any field $k$ (possibly of positive characteristic) and an odd degree rational map $\mathbb{...
- 5,958
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285
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Statistics for rational points on curves of genus $g$ over $\mathbb{F}_q$, $g\gg q$
Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot ...
- 5,958
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659
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Understanding a certain algebraic set arising in Deep Learning
I'm not a professional geometer. Thanks in advance for your patience.
So, let $n$, $k$, $p_0,\ldots,p_{k}$ be positive integers. Let $X$ (resp. $Y$) be an $p_0$-by-$n$ (resp. an $p_{k}$-by-$n$) real ...
- 6,853
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521
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CohFT: Witten vs. Kontsevich and Manin
Is there any connection to CohFTs as defined by Witten in his 1988 paper (via topological twist) and the CohFTs as defined by Kontsevich and Manin (in the context of Gromov-Witten theory of course).
...
- 661
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324
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Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack
I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.
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- 595
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257
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Curves on rational surfaces and Lang's conjecture for M_g
There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not ...
- 5,958
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592
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What is known about abelian varieties with several principal polarizations?
Let ``the simple case" be when the polarized abelian variety does not break up into a product of polarized abelian varieties.
I am trying to get an idea of what is known about abelian varieties with ...
- 3,539
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555
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Representation theory of finite groups with additional structures
Let $H$ be a finite group, representation theory of $H$ over $\Bbb C$ essentially determines $\operatorname{Hom}(H,GL_n(\Bbb C))$ up to conjugation action of $GL_n(\Bbb C)$ for each $n$. If we replace ...
- 6,622
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358
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Is the quotient of two linear group schemes linear?
Let $S$ be an affine scheme.
Call a group scheme $G\to S$ linear if there exists an $S$-group morphism $G\to \mathrm{GL}_{n,S}$ with trivial kernel.
Assuming this, suppose $H\to S$ is a central closed ...
- 2,928
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519
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Does Lefschetz-type theorems imply ampleness?
Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following properties:
$D$ is ample.
(Positivity) For any $k$-dimensional ...
- 1,170
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278
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Is there a geometric interpretation of the cohomology of an automorphism group acting on a universal deformation ring?
Let $X_0$ be some algebro-geometric object defined over a field, and suppose its deformation functor is prorepresentable, so there is a universal deformation ring $R$. Then $Aut(X_0)$ acts naturally ...
- 10.7k
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340
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Homology of Gersten complex for singular schemes
It is one of the important facts in K-theory/motivic cohomology that the Gersten-type complexes (for Quillen K-theory, Milnor K-theory or more generally Rost's cycle modules) are exact for smooth ...
- 17.4k
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273
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Has Kac's conjecture (*), from "Infinite root systems, representations of graphs and invariant theory", been proved?
Let $k$ be an algebraically closed field of characteristic zero, $V$ a finite dimensional $k$ vector space, $V^{\ast}$ the dual space, and $G$ an algebraic subgroup of $GL(V)$. Let $V_0$ be the points ...
- 156k
12
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811
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Number field analog of Artin-Tate $\Rightarrow$ BSD?
What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
user113452
12
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603
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Refinement of Hodge conjecture
This question deals with the classic Hodge conjecture on projective non-singular complex varieties, or in other words, projective Kähler manifolds. In Deligne's writeup for the Clay Foundation he says ...
- 35.5k
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726
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What is missing in the current constructions of pure and mixed motives?
Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
- 2,227
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265
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Galois groups of classical differential equations
I am currently on the lookout for good motivational examples for differential Galois theory, and I was wondering the following:
Is there a book or article devoted (either partially or completely) to ...
- 4,745
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283
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A local global principle over function fields
For each prime $p$, fix a place of $\bar{\mathbb{Q}}$ extending $p$ so we can reduce elements of $\bar{\mathbb{Q}}$ which are integral over $\mathbb Z$ and get elements of $\bar{\mathbb{F}}_p$. Now ...
- 30.5k
12
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623
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De Rham Cohomology in positive characteristic
This question is about the possible existence of a "compactified" algebraic De Rham cohomology in positive characteristic.
Namely, one knows that, for a smooth, but not proper, variety $U$ over a ...
- 121
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729
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Elkies' supersingularity theorem in higher dimension (in terms of the associated Newton polygon)
Elkies' supersingularity theorem: Given an elliptic curve $E$ over $\mathbb{Q}$, there are infinitely many primes $p$ such that $E$ is supersingular over $\mathbb{F}_p$.
I have seen another post on ...
- 3,539
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216
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Failure of surjectivity in Hotta-Springer specialization: examples for special unipotents?
Last weekend's workshop on Springer theory and its generalizations at UMass demonstrated how far the subject has expanded over four decades, but the original set-up for the Springer correspondence ...
- 52.9k
12
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862
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Intrinsic definition of the weight filtration
Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...
- 6,920
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731
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Product of a Schubert polynomial and a double Schubert polynomial
Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as
$$S_u(x)S_v(x;y)=\sum_w{c_{...
- 2,168
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1k
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Meaningful review of Moriwaki's "Arakelov Geometry"
I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry
book:
http://www.ams.org/bookstore-getitem/item=mmono-244
I could do the review the standard way in a day or ...
12
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1k
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Use of derivators to the theory of motives?
This is a rather imprecise question but i think this could become a interesting pool of ideas and comments.
The theory of motives has evolved to a complex field of research the moment Voevodsky (and ...
- 1,479
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1k
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How much algebraic geometry do I need to study complex geometry?
As one can deduce from the questions I have asked on MO, I'm interested in complex geometry. I am aware that there are many facets to the field, some of which I am more comfortable with than others. ...
- 19.3k
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624
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How tight is the Weil bound for this exponential sum?
Let $f$ be a nonconstant polynomial of degree $d$. Let $\psi$ be a character of the additive group of $\mathbb F_p$. By Weil, we have:
$$ \left| \sum_{x\in \mathbb F_p} \psi ( f(x)) \right| \leq (d-1)...
- 149k
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343
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Hodge Decompositions and Gamma Factors of Hasse--Weil L-Functions
Let $X$ be a projective variety over a number field $K$. If $\mathfrak{p}\unlhd\mathcal{O}_K$ is a prime ideal we can regard the Euler factor of its $m$th Hasse-Weil $L$-Function at $\mathfrak{p}$ as ...
- 643
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464
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Are automorphisms of abelian varieties detected by the formal group?
Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$.
Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with ...
- 12.1k
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580
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Cohomology and conifold transition for the quintic
Let $Y\subset \mathbb{C}P^4$ be the quintic threefold given by the equation $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4+5X_0X_1X_2X_3X_4=0$$
it has 125 singular points whose links are homeomorphic to $S^2\times S^...
- 9,870
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1k
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Hrushovski's Construction
Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial (...
- 1,872
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569
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Embedding of the product of two Grassmannians into a Grassmannian
Consider an embedding $$\Phi: G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})\rightarrow G_k(R^n)$$ of the product of two Grassmannians $G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})$ into $G_k(R^n)$, where $G_k(...
- 121
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2k
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Recommendation textbooks on D-module
I am going to take part in a seminar on D-modules and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, A Primer of Algebraic D-Modules
...
12
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891
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What is the intuition for $\mathbb{Q}^{ab}$ having cohomological dimension $1$?
I frequently talk to people who think of finite fields as arithmetic analogs of punctured discs. This makes some sense: the absolute Galois group of a finite field is the profinite completion of $\...
12
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453
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Where is the representability of the moduli of curves with framed points proved?
There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...
- 45.7k
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1k
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What is $M_g$ over a finite field, really?
Let $M_{g} \to \mathbb{Z}$ be the coarse moduli scheme of non-singular genus g curves over $\mathbb{Z}$. That is, suppose that $M_{g}$ co-represents the functor $M^{\sharp}_{g} : \text{Sch} \to \text{...
- 3,284
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2k
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Geometric points in geometry agree with "geometric points" of the Zariski topos
There is a general notion of a geometric point in topos theory. A geometric point is a geometric morphism $Set\to T$.
There is also a notion of a geometric point in algebraic geometry. A ...
- 19.6k
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530
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A commutative monoid associated with a finite abelian group
Let $M$ be a finite abelian group, and denote by $e_m$, for $m \in M$, the canonical basis of $\mathbb{Z}^M$. For $m, n \in M$ define elements $v_{m,n} \in \mathbb{Z}^M/\langle e_0\rangle$ as
$$
v_{m,...
- 181
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990
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Stacks in the fpqc topology
This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and ...
- 2,083
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440
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K-Weil cohomology theories?
I don't know very much about this stuff, so I'm a bit afraid that I'm being naive or stupid, and I apologize if I am --- but it seems to me that Weil cohomology theories, or at least the standard ...
- 21k