ac.commutative algebra - 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 fact that toric ideals are binomial prime ideals. There ne coulduse that the coordinate rings are characterized as those admitting a finegrading by an affine semigroup , i.e. presented by a binomial prime ideal (Prop. 1.11 in Eisenbud/Sturmfels "Binomial ideals").This question resulted from an Example that I discussed with MateuszMichalek. The examp...Read more

ag.algebraic geometry - Description of a birational map, Fulton's "Introduction to toric varieties"

This is an exercise of "Introduction to toric varieties" by Fulton, page 71, section 3.4, (the last one in this page.)The problem is initiated by constructing a complete (toric) variety which is not projective, by taking the fan $\Delta$ whose edges in $\mathbb{Z}^3$ are passing through $v_1=-e_1, v_2=-e_2, v_3=-e_3, v_4=e_1+e_2+e_3, v_5=v_3+v_4, v_6=v_1+v_4$ and $v_7=v_2+v_4$ where $e_i$'s are the standard basis of $\mathbb{Z}^3,$ and with cones through the faces of the triangulated tetrahedron shown, here.It can be shown that no strictly conv...Read more

toric varieties - Recommendations for binomial system solver

I am interested in solving binomial systems of the form$$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\ \vdots &\vdots \\\\ a_m x_1^{d_{m1}} x_2^{d_{m2}} \cdots x_n^{d_{mn}} + b_m x_1^{d_{m1}} x_2^{d_{m2}} \cdots x_n^{d_{mn}} &= 0 \end{cases}$$where the exponents may be negative. I.e., each equation has exactly two terms. For example$$ \begin{cases} 3 x_1^{2} x_2^{-5} + 4 x_1^{-1} x_2^{6} &= 0 \\\\ 2 x_1^{3} x_2^{5} - 7 x_1...Read more

When does a discrepant toric resolution induce a crepant resolution of a subvariety?

Let $Y$ be a complete intersection in a complete simplicial toric variety $X_\Sigma$ such that $\DeclareMathOperator{Sing}{Sing}\Sing(Y)\subset\Sing(X_\Sigma)$. Suppose that $\phi:X_{\widehat{\Sigma}}\to X_\Sigma$ is a toric resolution induced by a refinement $\widehat{\Sigma}$ of the fan $\Sigma$. Is there a simple way to check if the restriction of $\phi$ to $\widehat{Y}=\phi^{-1}(Y)$ induces a crepant morphism $\widehat{Y}\to Y$?...Read more

ac.commutative algebra - Do subsets of generators of a toric ideal generate a toric ideal?

Given a toric ideal, say $J$, in a polynomial ring $k[x_1,...,x_n]$ we can find a finite generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give a finite minimal generating set for $J$ such that every subset of generators also generates a toric ideal. If not, are there any known counterexamples in the general case?If yes, could you provide a reference? Does it generalize to lattice ideals?Motivation: For particularly chosen, generating sets of toric ideals there exist subsets that also gener...Read more

ag.algebraic geometry - Toric Fano manifolds with Picard number 1

As far as I know, toric Fano manifolds are classified only up to dimension 4. In dimension one the projective line is the only example. In dimension two we have five examples: $\mathbb P ^2$, $\mathbb P^1 \times \mathbb P^1$ and $\mathbb P^2$ blown-up at 1,2 or 3 points in general position.Toric Fano 3-folds were classified by K. Watanabe and M. Watanabe. Their result asserts that there are 18 types of such a manifold and that all of them have Picard number $\rho \leq 5$, the projective space $\mathbb P^3$ being the only one with $\rho =1$.In f...Read more

Derived categories of toric varieties and convex geometry

Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example).One of the most important objects that are associated to an algebraic variety is its derived category. So I'm wondering: are there any constructions or properties in convex geometry that are reflected in the derived category of associated toric varieties?...Read more

ag.algebraic geometry - 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,Gorenstein, toric variety. Is there a simple algorithm for computing the ring of global sections of the structure sheaf on X_{\sigma}? I would like the presentation in terms of generators and relations if possible. If this is not possible in general, I would be happy if this is possible under some nice, fairly general situation. Thank you for your ...Read more

toric varieties - Secondary fans and Stanley Reisner ideals

Consider a collection of points $S \subset \mathbb R^d$. I would like to understand all possible fans $\Sigma$ whose support is the cone over $S$: $|\Sigma| = cone(S)$. I have heard that the secondary fan does this for me, but I am having trouble parsing the relevant sections of GKZ. I would like to understand this completely, but to begin I would really like to know How to construct the resulting set of Stanley-Reisner ideals for each possible $\Sigma$.Can someone explain this and/or give a readable explanation of How the secondary fan i...Read more

toric varieties - 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 $\frac{1 - abcd}{(1-ac)(1-ad)(1-bc)(1-bd)}$Is there a software package that can compute multigraded Hilbert series? Can it be computed using Macaulay2?Alternatively, is there software that can compute the multigraded Hilbert series of a toric variety, specified by its fan?For this example$v_1 = (0,0,1), v_2 = (1,0,1), v_3 = (1,1,1), v_4 = (0,1,1)$ specify the vertices of ...Read more

ag.algebraic geometry - nef Cone of a Toric Variety

Is there a systematic/standard way of extracting the nef cone of a toric variety $X$ from its fan (or polytope)? Can I tell from the basis of $A^1(X)$ induced by the one-skeleton, based on coefficients, when a given divisor class is nef?In particular I am working working with blowups of $\mathbb{P}^n$. I am uncertain if that extra piece of information helps.I imagine that the answer to my question is yes, but I havent yet found a place where this is cleanly articulated. Thanks in advance....Read more