The toric Hilbert scheme of a lattice L in Z^n is the multigraded Hilbert scheme parameterizing all ideals in k[x_1,...,x_n] with Hilbert function value one for every degree in the grading monoid N^n/L. In this paper we show that if L is two-dimensional, then the toric Hilbert scheme of L is smooth and irreducible. This result is false for lattices of dimension three and higher as the toric Hilbert scheme of a rank three lattice can be reducible.
The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P^4 whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of t…
We want to understand the geometry of all irreducible components of the toric Hilbert scheme. Until now it is known that the coherent component is (up to normalisation) the toric variety associated to the state polytope of the toric ideal. For the non-coherent components it was only known that there exists such a polytope describing the normalisation. Using the local equations and various facts about toric Hilbert schemes, we will derive an explicit construction of the polytope corresponding to the normalisation of the underlying reduced structure of a given non-coherent component of the toric…
We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme of points on $\mathbb C^2$. We obtain an upper bound for the Newton-Okounkov body of the Hilbert scheme of points on any smooth toric surface. We conjecture that this upper bound coincides with the exact Newton-Okounkov body for the Hilbert schemes of points on $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, and Hirzebruch surfaces. These results imply upper bounds for the effective cones of these Hilbert schemes, which are also conjecturally sharp in the above cases.
The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vectorspace. All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component. However, it is unknown whether toric Hilbert schemes are always connected. In this chapter we illustrate the use of Macaulay 2 for exploring the structure of toric Hilbert schemes. In the process we will encounter algorithms from commutative algebra, algebraic geometry, polyhedral theory and geometric combinatorics.
We construct small (50 and 26 points, respectively) point sets in dimension 5 whose graphs of triangulations are not connected. These examples improve our construction in J. Amer. Math. Soc., 13:3 (2000), 611--637 not only in size, but also in that their toric Hilbert schemes are not connected either, a question left open in that article. Additionally, the point sets can easily be put into convex position, providing examples of 5-dimensional polytopes with non-connected graph of triangulations.
We show that every 3-dimensional affine normal quasihomogeneous $SL(2)$-variety has an equivariant resolution of singularities given by an invariant Hilbert scheme. This article treats the case where such $SL(2)$-variety is toric. The non-toric case is considered in the forthcoming article [Kub18].
Let $X$ be a complete $n$-dimensional simplicial toric variety with homogeneous coordinate ring $S$. We study the multigraded Hilbert function $H_Y$ of reduced $0$-dimensional subschemes $Y$ in $X$. We provide explicit formulas and prove non-decreasing and stabilization properties of $H_Y$ when $Y$ is a $0$-dimensional complete intersection in $X$. We apply our results to computing the dimension of some evaluation codes on $0$-dimensional complete intersection in simplicial toric varieties.
This article is a continuation of [Kub18], which proves that if a $3$-dimensional affine normal quasihomogeneous $SL(2)$-variety $E$ is toric, then it has an equivariant resolution of singularities given by an invariant Hilbert scheme $\mathcal H$. In this article, we consider the case where $E$ is non-toric and show that the Hilbert-Chow morphism $γ: \mathcal H \to E$ is a resolution of singularities and that $\mathcal H$ is isomorphic to the minimal resolution of a weighted blow-up of $E$.
For $m\in \IN, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a toric surface $S.$ For $m$ big enough, we connect the number of a class of these irreducible components to the number of exceptional divisors on the minimal resolution of $S.$
For an arbitrary field K, let I be an ideal in the ring K[[x,y]] expressible as a polynomial in either the pair of ideals (x, y^4) and (x,y) or the pair (x,y^3) and (x^2, y). Let G be the group of automorphisms of K[[x,y]] sending the ideal (x,y^2) to itself. Then the normalization of the closure of the G orbit of I is a toric variety. We use the interplay of the properties of Hilbert schemes and toric varieties to study these spaces.
For fixed degree $d\leq 12$, we study the Hilbert scheme of degree $d$ smooth Fano threefolds in their anticanonical embeddings. We use this to classify all possible degenerations of these varieties to toric Fano varieties with at most canonical Gorenstein singularities.
Let X be a smooth projective toric surface, and H^d(X) the Hilbert scheme parametrising the length d zero-dimensional subschemes of X. We compute the rational Chow ring A^*(H^d(X))\_Q. More precisely, if T is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring A\_T^*(H^d(X))\_Q and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces such as the affine plane are described by our method too.
Let \X be an affine toric variety under a torus \T and let T be a subtorus. The general T-orbit closures in \X and their flat limits are parametrized by the main component H_0 of the toric Hilbert scheme. Further, the quotient torus \T/T acts on H_0 with a dense orbit. We describe the fan of this toric variety; this leads us to an integral analogue of the fiber polytope of Billera and Sturmfels. We also describe the relation of H_0 to the main component of the inverse limit of GIT quotients of \X by T.
We define phylogenetic projective toric model of a trivalent graph as a generalization of a binary symmetric model of a trivalent phylogenetic tree. Generators of the pro- jective coordinate ring of the models of graphs with one cycle are explicitly described. The phylogenetic models of graphs with the same topological invariants are deforma- tion equivalent and share the same Hilbert function. We also provide an algorithm to compute the Hilbert function.
Let X be an irreducible affine T-variety. We consider families of affine stable toric T-varieties over X and give a description of the corresponding moduli space as the quotient stack of an open subscheme in a certain toric Hilbert scheme under the action of a torus.
For a toric pair $(X, D)$, where $X$ is a projective toric variety of dimension $d-1\geq 1$ and $D$ is a very ample $T$-Cartier divisor, we show that the Hilbert-Kunz density function $HKd(X, D)(λ)$ is the $d-1$ dimensional volume of ${\overline {\mathcal P}}_D \cap \{z= λ\}$, where ${\overline {\mathcal P}}_D\subset {\mathbb R}^d$ is a compact $d$-dimensional set (which is a finite union of convex polytopes). We also show that, for $k\geq 1$, the function $HKd(X, kD)$ can be replaced by another compactly supported continuous function $φ_{kD}$ which is `linear in $k$'. This gives the formula f…
For $m\in \mathbb{N}, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a normal toric surface $S.$ We give formulas for the number of these components and their dimensions. When $m$ varies, these components give rise to projective systems, to which we associate a weighted graph. We prove that the data of this graph is equivalent to the data of the analytical type of $S.$ Besides, we classify these irreducible components by an integer invariant that we call index of speciality. We prove that for $m$ large enough, the set of components with index of speciality $1,$ i…
In this manuscript we sharpen the lower bound on the codimension of the irreducible components of the Noether-Lefschetz locus of surfaces in projective toric threefolds given in [BG17]. We also provide a simpler proof of Theorem 4.11 in [BG17], which allows one to avoid some technical assumptions.
In this article, we investigate deformations of a Calabi-Yau manifold $Z$ in a toric variety $F$, possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor $H^F_Z$ of infinitesimal deformations of $Z$ in $F$ to the functor of infinitesimal deformations of $Z$ is smooth. This implies the smoothness of $H^F_Z $ at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.
We present an explicit formula for computing toric residues as a quotient of two determinants, a la Macaulay, where the numerator is a minor of the denominator. We also give an irreducible representation of toric residues by extending the theory of subresultants to monomials of critical degree in the homogeneous coordinate ring of the corresponding toric variety.
In this paper we compute the Hilbert functions of irreducible (or smooth) and reduced arithmetically Gorenstein schemes that are twisted anti-canonical divisors on arithmetically Cohen-Macaulay schemes. We also prove some folklore results characterizing the Hilbert functions of irreducible standard determinantal schemes, and we use them to produce a new class of functions that occur as Hilbert functions of irreducible (or smooth) and reduced arithmetically Gorenstein schemes in any codimension.
We study holomorphic foliations of aribitrary codimension in smooth complete toric varieties. We show that split foliations are stable if some good behaviour of their singular set is provided. As an application of these results, we exhibit irreducible components of the space of foliations that arise as pullbacks of some special T-invariant divisors.
We prove that the Hilbert scheme of 11 points on a smooth threefold is irreducible. In the course of the proof, we present several known and new techniques for producing curves on the Hilbert scheme.
The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric Calabi-Yau 3-folds. This is obtained by truncating the $-1$-shifted symplectic structure on the derived moduli stack [BBBBJ15]. In this paper we show the canonical d-critical locus structure has critical charts consistent with the description of Hilbert scheme as a degeneracy locus [BBS13]. In particular, the canonical d-critical locus structure is isomorphic to th…
We give detailed descriptions of the period maps of two 2-parameter families of anti-canonical hypersurfaces in toric 3-folds. One of them is related to a Hilbert modular surface, and the other is related to the product of modular curves.
We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space \bar{M}_{0,n} of stable genus zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.
We prove the existence of a compactly supported, continuous (except at finitely many points) function $g_{I, {\bf m}}: [0, \infty)\longrightarrow \mathbb{R}$ for all monomial prime ideals $I$ of $R$ of height one where $(R, {\bf m})$ is the homogeneous coordinate ring associated to a projectively normal toric pair $(X, D)$, such that $$\int_{0}^{\infty}g_{I, {\bf m}}(λ)dλ=β(I, {\bf m}),$$ where $β(I, {\bf m})$ is the second coefficient of the Hilbert-Kunz function of $I$ with respect to the maximal ideal ${\bf m}$, as proved by Huneke-McDermott-Monsky \cite{HMM2004}. Using the above result, fo…
The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is ZZ[x,y], which establishes a conjecture of Haiman and Sturmfels.
The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf $Ω_¶^p(D)$ of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope for ${Ø_X}(D)$. We also introduce a new combinatorial object, the so-called p-th Hilbert-Erhart polynomial, which generalizes the usual notion and behaves similar. Namely, there exists a generalization of the inversion law for a usual …
Let $X_\mathcal{A}$ be the projective toric variety corresponding to a finite set of lattice points $\mathcal{A}$. We show that irreducible components of the Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ parametrizing $k$-dimensional linear subspaces of $X_\mathcal{A}$ are in bijection to so-called maximal Cayley structures for $\mathcal{A}$. We explicitly describe these irreducible components and their intersection behaviour, characterize when $\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$ is smooth in dimension $k$, then every component of $\mathbf{F}_k(X_\mathcal{…
Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every "minimal embedded toric resolution". We prove that this correspondence is bijective except for the $E_8$ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.
We use cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.
Let H_{ab} be the equivariant Hilbert scheme parametrizing the 0-dimensional subschemes of the affine plane invariant under the natural action of the one-dimensional torus T_{ab}:={(t^{-b},t^a), t\in k^*}. We compute the irreducible components of H_{ab}: they are in one-one correspondence with a set of Hilbert functions. As a by-product of the proof, we give new proofs of results by Ellingsrud and Stromme, namely the main lemma of the computation of the Betti numbers of the Hilbert scheme H^l parametrizing the 0-dimensional subschemes of the affine plane of length l and a description of Bialyn…
Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that are numerically equivalent to a multiple of C. When X is projective, we compare contractible and extremal curves, and we show that every curve in X is numerically equivalent to a linear combination with positive integral coefficients of contractible curves.
We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting t…
In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the correspondence between the irreducible invariant difference subvarieties and the faces of the N[x]-submodules and the orbit-face correspondence. Finally, an algorithm is given to decide whether a binomial di…
Monomial ideals and toric rings are closely related. By consider a Grobner basis we can always associated to any ideal $I$ in a polynomial ring a monomial ideal ${\rm in}_\prec I$, in some special situations the monomial ideal ${\rm in}_\prec I$ is square free. On the other hand given any monomial ideal $I$ of a polynomial ring $S$, we can define the toric $K[I]\subset S$. In this paper we will study toric rings defined by Segre embeddings, we will prove that their $h-$ vectors coincides with the so called Simon Newcomb number's in probabilities and combinatorics. We solve the original questio…
Exceptional loci of F-blowups and $G$-Hilbert schemes
Instituto de Ingeniería y Tecnología, UACJ（乌斯帕塔技术工程研究所）；The University of Osaka（大阪大学）；Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo（东京大学宇宙物理与数学研究所）
We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or their irreducible components are smooth. In particular, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the Borel-fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel-fixed points.
We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer's conjecture on equations defining Grothendieck's classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes.
In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leqρ_X-ρ_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $ρ_X-ρ_D>0$ (with some additional hypotheses if $ρ_X-ρ_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $ρ_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension …
In this paper we show that the Hilbert scheme $H(3,g)$ of locally Cohen-Macaulay curves in $\Pthree$ of degree three and genus $g$ is connected. In contrast to $H(2,g)$, which is irreducible, $H(3,g)$ generally has many irreducible components (roughly $-g/3$ of them). To show connectedness, we classify the curves (giving particular attention to the triple lines), determine the irreducible components, and give flat families over $\Aone$ to show that the components meet. As a byproduct, we find that there are curves which lie in the closure of each irreducible component.
We study Fano schemes $F_k(X)$ for complete intersections $X$ in a projective toric variety $Y\subset \mathbb{P}^n$. Our strategy is to decompose $F_k(X)$ into closed subschemes based on the irreducible decomposition of $F_k(Y)$ as studied by Ilten and Zotine. We define the expected dimension for these subschemes, which always gives a lower bound on the actual dimension. Under additional assumptions, we show that these subschemes are non-empty and smooth of the expected dimension. Using tools from intersection theory, we can apply these results to count the number of linear subspaces in $X$ wh…
