## 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?