All Questions
15 questions
16
votes
3
answers
2k
views
The Sylvester Gallai Theorem and Sections of Varieties with "Simple Topology".
The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points.
One high dimensional extension ...
11
votes
2
answers
910
views
Access to a preprint by D. N. Verma
Some work I am doing is connected with a sequence 1, 3, 40, 1225, 67956, $\dots$ which agrees with http://oeis.org/A012250 for all eight terms. The only useful information in OEIS on this sequence is ...
- 50.8k
8
votes
1
answer
381
views
A "polar dual" for projective varieties?
Given a projective variety $X$ (over $\mathbb{C}$, say) with an affine paving $X=\sqcup_i C_i$, one can construct a poset $P_X$ on the set of cells $\{C_i\}$ by saying $C_i \leq C_j$ whenever $C_i \...
- 2,753
4
votes
0
answers
255
views
Economic equilibrium and tropical geometry
There is a famous saying in economics: When everyone pursues his or her own interests, there is an invisible hand that brings the market to equilibrium. However, this is not always the case. Here is ...
- 123
4
votes
0
answers
168
views
A fast way to test whether a partial function can be extended to a chirotope of rank 3 ?
Hello,
What is a fast way to test whether a partial function can be extended to a chirotope of rank 3 ? That is: I have a domain
$E = \{1,...,n\}$
and a partial function
$f: E^3 \to \{-1, 0, 1\}$
...
- 191
3
votes
0
answers
93
views
Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
- 183
3
votes
0
answers
60
views
Points of a centrally symmetric lattice polytope
Let $P\subseteq\mathbb R^n$ be a centrally symmetric lattice polytope whose only interior lattice point is the zero vector. Is it true that $P$ is equivalent (up to ${\rm GL}(n,\mathbb Z)$ + lattice ...
- 351
3
votes
0
answers
83
views
A weaker version of Randell Isotopy Theorem
I am studying a problem in hyperplane arrangement theory related to the homotopy type of the complement manifold of a certain class of hyperplane arrangements.
In a well celebrated paper Richard ...
- 741
3
votes
0
answers
137
views
What are the Voronoi cones in 4 variables?
Question: What are the top dimensional cones of the 2nd Voronoi decomposition of the space of positive definite forms in $4$ variables?
The 2nd Voronoi decomposition of the cone of positive definite ...
- 31
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
1
vote
0
answers
513
views
Maximal disjoint hyperplanes
Assume a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ is given occupying a codimension $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these ...
- 800
0
votes
1
answer
367
views
On the number of lines of given points
Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions :
I suppose Beck's theorem doesn't hold when instead ...
- 1
0
votes
0
answers
193
views
What breaks down in the theory of affine hyperplane arrangments?
It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and ...
- 101
0
votes
0
answers
230
views
Toric morphism fiber and kernel dimensions
Given a morphism between two smooth toric varieties $f: X \rightarrow Y$, is the dimension of the kernel of $\mathrm{d}f$ at any point $p \in X$ equal to the dimension of the fiber at $f(p) \in Y$?
...
- 1,719