# Tagged Questions

**4**

votes

**1**answer

222 views

### 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 ...

**4**

votes

**1**answer

193 views

### Continued fraction expansion of an algebraic number and its conjugates

Let $w$ be an element of a Galois extension $L:\mathbb{Q}$ such that $\text{Gal}(L/\mathbb{Q})=\langle g\rangle$ is cyclic of order $n$ (here $\mathbb{Q}$ is rationals). Suppose we know the continued ...

**1**

vote

**0**answers

151 views

### Number of Orbits of symmetric group acting on $(\mathbb{Z}/n)^{l}$ [migrated]

I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related ...

**4**

votes

**1**answer

121 views

### Transalate of a Richardson Variety

For $v \leq w$ are Weyl group elements, the intersection of a Schubert variety $X^w$ and the opposite Schubert variety $X_v$ is called a Richardson variety. It is denoted by $X_v^w$. It is well know ...

**2**

votes

**2**answers

230 views

### convex polytope integer points

is there a simple proof for the following lemma:
An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.

**0**

votes

**0**answers

59 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 $f(p) \in Y$?
...

**2**

votes

**3**answers

166 views

### branching schubert calculus

Let $X=Gr(r,V), Y=Gr(r+1,W)$ where $V,W$ are complex vector spaces with $\dim V > r$ and $\dim W > \dim V$. Let $\phi:X\rightarrow Y$ be some embedding of varieties. This induces a morphism on ...

**2**

votes

**0**answers

257 views

### 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, ...

**5**

votes

**1**answer

239 views

### Grassmann-Plücker relations for permanents

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann-Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of ...

**2**

votes

**1**answer

78 views

### Point-Hyperplane incidence in finite projective spaces

Let $P$ be a finite projective space of order $q$ and dimension $d$. I am interested in finding the least $k$ such that for any set $S$ of $k$ points of $P$, and for any set $S'$ of $k$ hyperplanes of ...

**3**

votes

**0**answers

116 views

### What is the combinatorial data classifying non-normal affine toric varieties?

Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...

**4**

votes

**2**answers

203 views

### Is there a theory of oriented subspace arrangements?

The theory of hyperplane arrangements is a rich and intensely studied subject, especially from the perspective of combinatorics; see e.g. this wonderful monograph of Stanley. Oriented hyperplane ...

**6**

votes

**0**answers

191 views

### Fomin-Kirillov algebras and Schubert calculus

In
Fomin, Sergey; Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and
Schubert calculus. Advances in geometry, 147--182, Progr. Math., 172,
Birkhäuser Boston, Boston, MA, 1999. MR1667680 ...

**0**

votes

**0**answers

114 views

### Intersection theory on moduli spaces of curves without marked points

1. There are a lot of works concerning the intersection theory on the moduli spaces of curves $\mathcal M_{g,n}$ (and their Deligne-Mumford compactifications $\overline{\mathcal M}_g$), for $n>0$.
...

**3**

votes

**2**answers

197 views

### Kahler differentials on cluster varieties

On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...

**0**

votes

**0**answers

48 views

### Finding stable ideals of F_3[[X,S]] by group action.

Let k > 1 be a positive integer and define the action σ_k on F_3[[X,S]] by
σ_k: X ---> X + S + X^k
σ_k: S ---> S + S^3.
Then,
Conjecture: There exists a principal ideal (a) other than (S) such ...

**3**

votes

**2**answers

251 views

### computing monodromy of branched cover of sphere specified by a polynomial

We know by the Riemann Existence Theorem that any Riemann surface can arise holmorphically as the branched cover of a sphere:
Which Riemann surfaces arise from the Riemann existence theorem?
...

**13**

votes

**2**answers

1k views

### What is Tropicalization, and how is it applied

My question is:
What is Tropicalization, how is it done, and what are some basic applications of it?
motivation
I am interested especially in how questions about enumerative algebraic geometry ...

**9**

votes

**0**answers

175 views

### Degree of a cone over the set of rank $r$ $n\times n$ matrices

Let $X_r\subset Mat_{n\times n}(C)$ denote the variety of rank at most $r$ matrices, set $k=n-r$ and assume $n\geq k^2-1$. Consider the cone over $X_r$ with vertex spanned by the first $k^2-1$ entries ...

**11**

votes

**1**answer

497 views

### Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...

**2**

votes

**0**answers

158 views

### Schemes defined by a collection of Plücker coordinates

If $C \subset {[n]\choose k}$ is any collection of $k$-element sets, we can define a scheme $$ W(C) = \bigcap_{S\notin C} \{V \in Gr(k,n) : p_S(V)=0\} \qquad \subseteq Gr(k,n), $$ where $p_S$ is the ...

**2**

votes

**1**answer

280 views

### Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they ...

**3**

votes

**0**answers

103 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 ...

**9**

votes

**4**answers

526 views

### zeros of a homogeneous polynomial

Hi All,
Let $F$ be a finite field, $\lambda\in F$, and $$p_\lambda (x,y,z)=\left|\begin{array}{ccc}x & y & z \\ y & z & x +\lambda z \\ z& x+\lambda z & y+\lambda x+\lambda ...

**2**

votes

**0**answers

156 views

### global sections of structure sheaf on the toric Calabi-Yau

Let P be a lattice polytope and lying in $ N \times {1} \subset N \times \mathbb{R}$. Let $\sigma$ be the cone over this polytope and $X_\sigma$ be the corresponding toric variety, which is an ...

**0**

votes

**1**answer

49 views

### Connection between the number of vertices and the number of lattice points of the integer hull of a polytope?

Is there a connections between the number of vertices and the number of lattice points of $P_I$, the integer hull of a polytope $P$? Which is usually more difficult to determine?
Or if I have a bound ...

**10**

votes

**1**answer

725 views

### Smith Normal Form of powers of a matrix

What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...

**4**

votes

**3**answers

178 views

### Generate a higher degree symmetric polynomial from an existing one

Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as
$q(x_1, x_2, \cdots, x_{n+1}) = ...

**0**

votes

**3**answers

215 views

### When does the rigidity matrix of a graph have full row rank?

Intuitive description: In the 2D plane, there are $m$ bars connected by $n$ joints. The length of each bar is fixed. These joints and bars can be viewed as a graph (see the figures below). Denote ...

**6**

votes

**3**answers

410 views

### What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?

(From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
...

**3**

votes

**0**answers

292 views

### Tropicalization of the Grassmannian

Let $Trop(Gr(m,n))$ denote the tropicalization of the grassmannian $Gr(n,m)$. Let $\phi^m : \mathbb R^{n \choose 2} \rightarrow \mathbb R^{n \choose m}$ such that $X_{i,j} \rightarrow ...

**16**

votes

**2**answers

413 views

### A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...

**3**

votes

**2**answers

454 views

### Reference request: Lascoux's formulas for Chern classes of tensor products and symmetric powers

Let $E$ and $F$ be vector bundles on a smooth projective variety, say.
A. Lascoux ("Classes de Chern d'un produit tensoriel", C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 8, A385–A387) gave ...

**1**

vote

**0**answers

51 views

### Combinatorical surface is restricted to a closed face an injection

Hello :)
I'm third year student of mathematics. In my own intrest i'm studying topology in combinatorical sense. Herefore i found also an lecture note in knot theory from Roberts. I want to understand ...

**11**

votes

**2**answers

704 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 ...

**4**

votes

**1**answer

174 views

### Family of hypersurfaces in (C^*)^2 corresponding to tropical family

Edit: I realize the mathematics below is lacking a precise phrasing. I hope that the intuitiion behind the question is clear enough that a reader will understand the question and provide guidance. The ...

**4**

votes

**2**answers

396 views

### Systems of simultaneous real quadratic equations

Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of ...

**5**

votes

**0**answers

281 views

### Tensor rank of anti-symmetric tensor

Let $V$ be a vector space of dimension $n$. Let us consider $V^{\otimes n}=V\otimes V \ldots \otimes V$. This vector space contains one dimentional vector space $\wedge^n V$. My question is does it ...

**2**

votes

**2**answers

419 views

### elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C.
Let I={1,...,r} the set of simple roots.
For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...

**8**

votes

**0**answers

257 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 ...

**6**

votes

**2**answers

278 views

### Finding the matroids with a specified set of non-bases

I'm a grad student in algebraic geometry, and I've encountered a problem which requires me to produce an algorithm involving matroids. Since this isn't my area of expertise, I'm hoping someone knows ...

**14**

votes

**1**answer

451 views

### Smooth bases of matroids

Motivated by algebraic geometry, I've come up with a purely
combinatorial definition within the theory of matroids.
The question is: is this concept known?
If you like matroids but not algebraic ...

**17**

votes

**4**answers

673 views

### The Constructions of Davis and Januszkiewicz

One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...

**27**

votes

**2**answers

2k views

### Shimura-Taniyama-Weil VS Grothendieck's dessins

When listening to the beautiful lectures by Gilles Schaeffer at
the SLC68, the following (perhaps crazy) question occurred to me:
did anyone attempt (succeed?) to combinatorially prove modularity of ...

**0**

votes

**1**answer

338 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 ...

**9**

votes

**2**answers

455 views

### Detecting tilings by toric geometry

This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask.
Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...

**1**

vote

**1**answer

249 views

### Incidences of Lines / Circles in the Plane

During linear algebra class, I was explaining that given 2 equations + 2 unknowns where we expect there to be a unique solution but sometimes there can be 0 solutions or a line's worth.
At some ...

**3**

votes

**1**answer

323 views

### What is the Bruhat decomposition of the affine Grassmannian?

We define the affine Grassmannian to be the quotient $Gr = GL_n(\mathbb{C}((t)))/GL_n(\mathbb{C}[[t]])$ where $\mathbb{C}((t))$ is the field of formal Laurent series and $\mathbb{C}[[t]]$ is the ring ...

**29**

votes

**1**answer

1k views

### What is the sandpile torsor?

Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...

**0**

votes

**0**answers

112 views

### Non-regular (Non-coherent) subdivisions of a polygon.

There are many papers and books which study about the regular subdivision of a convex lattice polytope.
My question is about "Non"-regular subdivisions of a 2-dimensional convex lattice polygon.
I ...