All Questions
25 questions
3
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Intersection numbers of moduli spaces and noncrossing partitions
The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
- 10.5k
7
votes
1
answer
481
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Geometric foundation of the Grothendieck polynomials
Grothendieck polynomials were firstly defined in
Alain Lascoux and Marcel-Paul Sch¨utzenberger. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une vari´et´e de drapeaux....
- 27.8k
1
vote
0
answers
173
views
The geometry of a commutative ring and the topology of its ideal complex
Suppose $R$ is a commutative Noetherian ring. Let $\mathcal{P}(R)$ be the poset of ideals of $R$ ordered by inclusion, and let $\Delta(R)$ be the order complex of $\mathcal{P}(R)$. $\Delta(R)$ is a ...
- 19
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votes
1
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308
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About Alexander method in mapping class group
The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups"
For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...
- 28.2k
12
votes
3
answers
2k
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Mnev's universality corollaries, quantitative versions?
Mnev's universality theorem claims that any semialgebraic set is the realization space of some oriented matroid. Moreover, the rank of the or matroid can be prescribed in advance.
1.-Are there ...
- 8,842
1
vote
0
answers
231
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Cohomology of realization space of matroid
Do we know any thing about cohomology of realization space of matroid (the space of all set of vectors in $\mathbb{C}^k$ which captures the independence structure of matroid $M$), more simple, for ...
- 91
8
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313
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Cohomology of the complement of the resonance hyperplane arrangement
Here was a question about resonance arrangement. It is defined as follows.
Let $x_i$ be the standard coordinates on $\mathbb{C}^n$. For each nonempty $I\subseteq\{1,\dots,n\}$, define the hyperplane $...
- 743
9
votes
0
answers
361
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Bernoulli-like polynomials
Let $\psi_0 (x,t)=\frac{te^{xt}}{1-e^{-t}}$. Then
$$\psi_0(0,t)=\frac{t}{1-e^{-t}};$$
$$\psi_0(x,t)=1+\sum_{n=1}^\infty \frac{t^n}{n!} B_n(x)$$
where $B_n$ is a monic polynomial of degree $n.$
Now ...
- 401
15
votes
5
answers
2k
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Striking existence theorems with mild conditions, and simple to state: more recent examples?
I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild ...
- 14.7k
27
votes
2
answers
2k
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Combinatorics of K(Z,2)?
Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...
- 1,482
7
votes
1
answer
451
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Inequality number of facets simplicial complex
In a recent preprint, Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies a version of ...
- 732
20
votes
1
answer
902
views
Double Counting: Motivic Edition
One of the most important proof techniques in combinatorics is double counting: proving that both sides of an identity count elements of some set in two different ways. This question is an attempt at ...
- 24.2k
3
votes
1
answer
237
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Singularity of torus fixed points from combinatorial data
May I ask what are the relations between the geometry and combinatorics near a torus fixed point? Any references?
In particular, let $S$ be a scheme that is torus invariant with finitely many zero and ...
- 121
7
votes
1
answer
710
views
Bialynicki-Birula Decomposition and moment polytopes/graphs
Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/...
- 15.5k
2
votes
1
answer
306
views
connectedness of fibers of torus-equivariant moment maps
Given a possibly singular, connected, symplectic algebraic variety with a torus action, every fiber of the moment map admits a torus action. Is each fiber of this moment map connected? Any examples or ...
- 15.5k
5
votes
1
answer
733
views
To derive or not to derive, that is the question
What are concrete and abstract examples of problems (even whole programmes of inquiry) when one has a choice to use a "derived" theory (e.g., $\infty$-categories, DAG, HAG, $DRep_k(G)$, "higher" ...
- 4,111
2
votes
1
answer
577
views
The concept "opposite" of Cohen-Macaulayness
Let $S = \mathbb k[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb k$ and let $I \subseteq S$ be a monomial ideal.
For a monomial ideal $J$, let $\#(J)$ be the smallest number of ...
CommunityBot
- 1
0
votes
1
answer
390
views
Moment maps and flat degenerations of toric varieties
We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...
- 27.8k
1
vote
1
answer
207
views
Singular homology of the zero loci of polynomials
I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...
- 15.2k
1
vote
0
answers
54
views
Lattice-isotopic essentialization of arrangements
I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
- 741
5
votes
0
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604
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The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)
(Migrated from MSE)
While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
CommunityBot
- 1
5
votes
1
answer
312
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Non-vanishing of elements in cohomology of full Flag varieties
Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$:
$0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
- 156k
8
votes
0
answers
636
views
Chern Classes of Exterior Products of a vector bundle.
This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of $...
- 327
0
votes
1
answer
225
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Is the Euler characteristic of a certain nonlinear variety related to that of a certain linear variety?
(This is a generalization of a question I posted a week ago.)
I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + \...
CommunityBot
- 1
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0
answers
2k
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Ignore this question [closed]
This question is a hacky way to create some tags for you to use. Move along.
