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?

ag.algebraic geometry: special kinds of ideals (for example, toric) that support a faster Buchberger algorithm?

I have heard that the Toric ideals allow Buchberger's algorithm to be considerably accelerated (see Basics of Grobner's Toric ideals, Observation 2,3). My question is twofold:

  • What are the precise theoretical limits of complexity known for Buchberger's algorithm for toric ideals?

  • Are there other kinds of non-trivial ideals on which a grobner base can be calculated efficiently? Can you please provide a reference?

For context, I want to use a grobner base as a way to encode data flow analysis problems in the compiler's construction, so that it allows multiple analyzes to "share" information easily. Therefore, knowing special types of ideals and fast algorithms to calculate your Grobner base would help design specific data flow analysis.

ag algebraic geometry – equivalent cohomology algebra of toric variety

Leave $ X $ be a complex, projective and smooth toric variety of complex dimension $ n $. It is acted by the true bull $ T = (S ^ 1) ^ n $.

Is it true that the $ T $-cohomology equivalent $ H ^ * _ T (X, mathbb {Z}) $ from $ X $ is isomorphic as a graduated algebra for $$ H ^ * (X, mathbb {Z}) otimes H ^ * _ T (pt, mathbb {Z}) simeq H ^ * (X, mathbb {Z}) otimes mathbb {Z } (x_1, points, x_n), $$
where $ x_i $ have grade 2?

Observation. I think I have proof of such isomorphism in the graduate category $ H_T ^ * (pt, mathbb {Z}) $-modules instead of graduated algebras (which seems to be well known).

algebraic geometry ag. – Local toric varieties and tropicalization

Leave $ K $ be a valuable field and consider the ring $ R = K ((x_1, points, x_m)) $ from Laurent's formal series. This is "the germ of the bull in $ 0 $"Is there a theory of" local toric varieties "where $ R $ replace the usual ring $ K (x_1 ^ { pm 1}, dots, x_m ^ { pm 1}) $?

One observation: one easily sees that a necessary condition for a fan to define a local toric variety is that the fan support lies in $ ( mathbb {R} _ { geq 0}) ^ m $. This makes sense, since none of the coordinate functions can have poles near $ 0 $.

The next step would be to tropicalize a subset $ Y $ from $ text {Spec} (R) $, which should give a tropical variety $ Trop (Y) $ inside $ ( mathbb {R} _ { geq 0}) ^ m $. When $ K $ You are given the trivial assessment, $ Trop (Y) $ it must be a fan and any local toric variety whose fan is compatible with $ Trop (Y) $, should give a "compaction" of $ Y $.

It is not clear to me how this should work. Is there any literature on this subject?