All Questions
Tagged with toric-varieties ac.commutative-algebra
18
questions
2
votes
0
answers
97
views
Kouchnirenko's theorem for non-generic polynomials
In Polyèdres de Newton et nombres de Milnor (Theorem 1.18), Kouchnirenko proved that given Laurent polynomials $f_1, \dotsc, f_k$ in $k$ variables, the number of isolated solutions is less than or ...
- 315
7
votes
0
answers
163
views
Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?
[This is an updated version of https://math.stackexchange.com/questions/4522399/.]
Let $f = \sum_{i=0}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_0,f_n \ne 0$)...
- 71
0
votes
0
answers
70
views
Low rank approximation
Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ...
4
votes
1
answer
1k
views
Blow-ups of toric varieties
I'm interseted in blow-ups of toric varieties, unfortunately, I don't understand the construction of a blow-up built by a refinement of a fan, if to be more specific, I didn't find any constructions.
...
- 123
2
votes
0
answers
104
views
Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,690
4
votes
0
answers
224
views
Intersection numbers on blow ups of toric varieties
Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
- 357
2
votes
0
answers
128
views
Explicit equations for conormal bundle to an affine toric variety
Let $L \subset \mathbb{Z}^n$ be a lattice and let $X_L$ be the closed toric subvariety of $\mathbb{C}^n$ cut out by the lattice ideal $I_L = \{x^{l_+} - x^{l_-} \,| \, l_+, l_- \in \mathbb{N}^n \text{ ...
- 1,543
7
votes
1
answer
840
views
Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?
It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs ...
- 15.5k
0
votes
2
answers
406
views
Computing toric ideals via saturation and Groebner bases of toric ideals
About a month ago I asked this question on math.stackexchange and unfortunately there was no response. Perhaps someone here knows the answer.
Let $A \in \mathbb{Z}^{m \times n}$ be a matrix of full ...
- 103
4
votes
1
answer
194
views
Toric ideal of slice of a polytope?
Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...
- 882
5
votes
1
answer
524
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 affine,...
5
votes
3
answers
1k
views
Stanley-Reisner ring of a simplicial complex is a functor?
Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let $\...
- 1,569
10
votes
1
answer
806
views
Triangulations of polytopes and tilings of zonotopes
Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...
- 882
11
votes
1
answer
860
views
Proving that a variety is not (isomorphic to) a toric variety
Is there an algorithmic (or other) way to prove that a (projective)
variety is not isomorphic to a toric variety?
I'd be happy with an algebraic answer (for affine or projective varieties),
using the ...
- 1,961
9
votes
0
answers
349
views
Computing Ext for toric divisors
Ok, I have an affine (normal) toric variety $X = \text{Spec} k[\sigma]$. Suppose that $F, G$ are two torus invariant Weil divisors on $X$. Is there any relatively straightforward way to compute
$$
\...
- 20.4k