Questions tagged [toric-varieties]
Toric variety is embedding of algebraic tori.
321
questions
10
votes
1
answer
1k
views
Cox rings of toric varieties over arbitrary fields
The Cox ring of a toric variety X can be viewed as a generalisation of the homogeneous coordinate ring of projective n-space. Over the complex numbers, the theory is outlined in The Homogeneous ...
12
votes
0
answers
526
views
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
$$
v_{m,...
5
votes
1
answer
1k
views
Why does the Euler characteristic of a toric variety equal the number of vertices in the defining polytope?
In this link, Corollary 3.2.2, page 59 the author claims that: The Euler characteristic of the toric variety $X_K$ associated to a convex polytope $K$ is the number of vertices of $K$.
I want to see ...
8
votes
1
answer
1k
views
Software for computing multi-graded Hilbert series
The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{*}$ with weights $(1,1,-1,-1)$
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$$
\frac{1 - abcd}{(1-...
4
votes
2
answers
2k
views
Moment map for toric actions -- online references?
Consider a toric variety, defined as a (normal?) complex projective variety $X$ together with an algebraic action of $(\mathbb C^*)^n$ with finitely many orbits. Now we have two "real symplectic" ...
10
votes
3
answers
1k
views
Hamiltonian $S^1$ actions with isolated fixed points
I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian $S^1$-action with isolated fixed points, that does not admit a compatible $...