ag.algebraic geometry – How to create a toric variety whose Cox ring has a specific grading?

If one wanted to obtain a fan for a toric variety of dimension $ n>1 $ whose Cox ring is $ mathbb{Z}^{2} $ graded with weights $ {(a_{i},b_{i})}_{i=1}^{n+2} $, then one could let $ B $ be the $ (n+2)times 2 $ matrix whose $ i $-th row has entries $ (a_{i},b_{i}) $. After computing the nullspace, one ends up with an $ n times (n+2) $ matrix $ A $ with entries in $ mathbb{Z} $. The $ n+2 $ columns of $ A $ are the rays for a fan in $ mathbb{R}^{n} $. If the rays are $ {u_{rho_{1}},dots,u_{rho_{n+2}} } $, then maximal dimensional cones $ sigma $ are of the form $ operatorname{Cone}(u_{rho_{i_{1}}},dots,u_{rho_{i_{n}}}) $. The fan $ Sigma $ is then obtained from the maximal cones and their faces. From $ Sigma $ one obtains an ideal $ B(Sigma) = langle x^{widehat{sigma}} rangle_{sigma in Sigma} $ where $ x^{widehat{sigma}} $ is $ prod_{i mid rho_{i} notin sigma} x_{i} $. From here the quotient of $ mathbb{A}^{n+2}_{mathbb{C}} setminus Z(B(Sigma)) $ by the $ mathbb{G}_{m}^{2} $ action which sends $ x_{i} $ to $ z_{1}^{a_{i}}z_{2}^{b_{i}}x_{i} $ is isomorphic to the variety $ X_{Sigma} $ obtained from the fan $ Sigma $. As a result, the Cox ring of $ X_{Sigma} $ has the desired grading.

What if instead of wanting to find an explicit fan of a toric variety of dimension $ n>1 $ whose Cox ring is $ mathbb{Z}^{2} $ graded, one wants to find an explicit fan of a toric variety of dimension $ n>1 $ whose Cox ring is $ operatorname{Hom}(mathbb{Z}/langle M rangle mathbb{Z}, mathbb{C}^{ast}) times operatorname{Hom}(mathbb{Z}/langle N rangle mathbb{Z}, mathbb{C}^{ast}) $ graded with weights $ (overline{a_{i}}, overline{b_{i}})_{i=1}^{n} $? Is there a similar algorithm for obtaining the fan for such a variety?

ag.algebraic geometry – Integral isomorphism between $K_0(X)$ and $A(X)$ for toric varieties

Let $X$ be a smooth projective toric variety. The Chern character gives an isomorphism of rings:
$$operatorname{Ch}:K_{0}(X)otimesmathbb{Q} to A(X)otimes mathbb{Q} $$
where $K_{0}(X)$ is the Grothendieck group of vector bundles on $X$ and $A(X)$ is the Chow ring of $X$. This map seems only well-defined over $mathbb{Q}$, but I was wondering (likely naively) if there is possibly an integral isomorphism (i.e. without tensor with $mathbb{Q}$)?

Why might we hope for such a map? Fulton and Strumfels showed that there exists an isomorphism $mathcal{D}_{X}:A^{k}(X)to operatorname{Hom}(A_{K}(X),mathbb{Z})$ where $A^{k}(X)$ and $A_{k}(X)$ are the Chow cohomology and homology groups respectively. In particular, this means that the Chow ring $A(X)$ of a smooth toric variety is torsion free. In the couple very (very) simple examples I’ve done $K_{0}(X)$ also seems torsion free, although I am unsure whether this is true generally.

Of course, even if both $K_{0}(X)$ and $A(X)$ are torsion free there need not be an isomorphism between them, but one can hope.

When is a toric variety a Poincare duality space?

When is a complete toric variety a Poincare duality space? Is there an "if and only if" condition? And is this condition local? Given an analytically-locally-toric compactification of a smooth variety, can the condition be described purely in terms of the local cones?

ag.algebraic geometry – the map on divisor class groups induced by restriction to a toric subvariety

Let $X$ be a (say, complex) toric variety acted upon by a torus $T$ and defined by a fan $Sigma$ in the cocharacter lattice $N=mathrm{Hom}(mathbb{C}^times, T)$, and let $M$ be the character lattice. For any cone $sigma in Sigma$ put $M(sigma) = sigma^perp cap M$, $N(sigma) = mathrm{Hom}(M(sigma), mathbb{C}^times)$. There is a natural projection $N to N(sigma)$. Then the closure of the orbit corresponding to $sigma$ has the structure of a toric variety with respect to the quotient torus with the cocharacter lattice $N/N(sigma)$ and given by the fan $Star(sigma)$ consisting of the images in $N(sigma)$ of the cones of $Sigma$ containing $sigma$. Note that the closed embedding $X_{Star(sigma)} to X$ is generally not a toric morphism, since the dense toric orbit of $X_{Star(sigma)}$ does not intersect the dense toric orbit of $X$.

My question is: is there a way to describe the restriction map $mathrm{Cl}(X) to mathrm{Cl}(X_{Star(sigma)})$ in terms of the fans $Sigma$ and $Star(sigma)$?

ag.algebraic geometry – Hodge structure on intersection cohomology of toric varieties

Given a convex polytope with integer vertices, one can construct a complex projective variety $X$ called toric variety. In general $X$ is not smooth. As I have heard, by the work of M. Saito, the intersection cohomology of any projective complex variety, in particular of $X$, carries a natural pure Hodge structure.

Is it true that it satisfies
$$H^{p,q}=0 mbox{ if } pne q?$$

This is known to be the case if $X$ is smooth.

ag.algebraic geometry – Curve with no embedding in a toric surface

I am looking for a smooth proper curve $C$ such that there does not exist any closed embedding $C to S$ where $S$ is a (normal projective) toric surface.

Using the result on p.25 of Harris Mumford, On the kodaira dimension of the moduli space of curves, I can conclude that a very general curve cannot have any such embedding.

However, I am not able to write down an explicit example. Does anyone know such an example or what sort of obstruction might work to check this in particular examples.

ag.algebraic geometry – Gromov-Witten invariants of cocharacter closures in toric varieties

$require{AMScd}$

Let $X$ be a toric projective variety with dense algebraic torus $iota:(mathbb{C}^times)^n to X$, and let $u:mathbb{C}^times to X$ be a cocharacter, by which I mean a map admitting a factorization of the form
$$mathbb{C}^times xrightarrow{h} (mathbb{C}^times)^n xrightarrow{iota} X qquad text{where}qquad htext{ is a group homomorphism}$$

Definition. The closure $bar{u}:C to X$ of the cocharacter $u$ is the unique extension of $u$ to a singular toric curve $C$ that commutes with the $mathbb{C}^times$-action on $mathbb{C}^times$ and $C$.

This construction seems pretty natural to me. Furthermore, a cocharacters are abundant since a cocharacter $u$ is equivalent to an element of $mathbb{Z}^n$ via the map
$$a = (a_1,dots,a_n) mapsto u_a qquadtext{with}qquad u_a(z) = (z^{a_1},dots,z^{a_n})$$
However, I am having trouble finding information about these curves. For example, I am interested in the following question.

Question 1. Are there other characterizations of the curves arising from this construction?

I am also interested in the Gromov-Witten theory of these curves. All that I can ask here is the following vague question.

Question 2. Is there some sense in which the curves $bar{u}_a$ has a “non-trivial count in Gromov-Witten theory”?

I’m lookin for an answer like: for each $a in mathbb{Z}^n$, there exists a $0$-dimensional moduli space of stable curves $overline{mathcal{M}}_{g,n}(X,A)$ that naturally includes $bar{u}_a$ (somehow) and where $GW^{X,A}_{g,n} neq 0 in H_0(X)$. This is almost certainly too specific, but anything in this general direction would be great.

ag.algebraic geometry – Distinguishing ample divisors by minimally intersecting curves on a projective simplicial toric variety

My question has an easily formulated generalization, which I will state first. Let $sigma subseteq mathbf{R}^n$ be a strongly convex polyhedral cone. For each minimally generating lattice point $m in sigma^o cap mathbf{Z}^n$ of the interior cone $sigma^o subseteq sigma$, let $S(m) subseteq sigma^{vee} cap mathbf{Z}^n$ denote the set of lattice points $u$ with $langle u,m rangle = 1$. My question is:

Does $S(m) = S(m’)$ imply that $m = m’$?

As a special case, assume that $sigma$ is the nef cone of a simplicial projective toric variety $X_{Sigma}$. Then my question seems to amount to the following:

If $D_1$ and $D_2$ are two ample divisors minimally generating in the ample cone, then does $D_1 cdot C = 1 Leftrightarrow D_2 cdot C = 1$ for all effective curves $C$ imply that $D_1 = D_2$?

This is the case I am most interested in.

ag.algebraic geometry – Toric compactification of toric Calabi-Yau’s

Let $X$ be a toric Calabi-Yau in complex dimension $ngeq 3$. In particular, this means that it is described by some fan $F$ that is spanned by vectors lying in the hyperplane $H_1 = {(x_1,ldots,x_n) in mathbb{R}^n,:, sum^n_{i}x_i = 1}$. We also have that $X$ is smooth.

A toric spin compactification of $X$ is a smooth projective toric variety given by a fan $tilde{F}$ containing $F$, such that the added divisors at infinity correspond to vectors in the union of hyperplanes

$$H_{text{odd}} = bigcup_{kin 2mathbb{Z}+1}H_k,,qquad H_{k} = {(x_1,ldots,x_n)inmathbb{R}^n,:,sum^n_{i=1}x_i = k},.$$

Is it possible to construct an example of $X$ for which there doesn’t exist a toric spin compactification?

Algebraic geometry ag: Groebner base of an toric ideal

I know about the Toric ideals that is a kind of binomial ideal, that is, generated by $ x ^ u – x ^ v $, where $ Au = Av $ (A is the associated matrix). So, finding all the whole solutions of $ AX = 0 $Can we somehow find the generators of the toric ideal? More specifically, can we determine the basis of Grobner? With the help of whole solutions?