Hilbert Schemes and Toric Degenerations for Low Degree Fano Threefolds

The majority of irreducible components we are interested in come from smooth Fano threefolds as follows. If $V$ is a smooth Fano threefold with very ample anticanonical divisor, then the Hilbert polynomial of $V$ in its anticanonical embedding is determined solely by its degree $d=(-K_{V})^{3}$. We denote the Hilbert scheme parametrizing subvarieties of $\mathbb{P}(|-K_{V}|)$ with this Hilbert polynomial by $\mathcal{H}_{d}$. The variety $V$ together with the anticanonical embedding corresponds to a point $[V]\in\mathcal{H}_{d}$, and this point lies on a single irreducible component, cf. [MM86].

Deformation families of smooth Fano threefolds have been completely classified, see [Isk78] and [MM82]. Each family is distinguished by the degree $d$, the second and third Betti numbers $b_{2}$ and $b_{3}$, and the Lefschetz discriminant of any threefold in the family. For threefolds of degree less than $30$, the first three invariants suffice. The degree must always be even, and can range from $2$ to $64$. We shall refer to the families and general elements of the families interchangeably.

Using this classification, we do some calculations to determine exactly which Fano threefolds of degree $d\leq 12$ have very ample anticanonical divisor and thus give rise to a component of $\mathcal{H}_{d}$, see §3.2. The results are recorded in Table 1. Note that all varieties appearing in the table have index one, except for $V_{8}^{\prime}$, which has index $2$. If $-K_{V}$ is very ample, we record how many global sections the corresponding normal sheaf has, which we calculate using our Proposition 3.1. This is just the dimension of the corresponding component of $\mathcal{H}_{d}$, to which we also give a name in the table.

In addition to the components of Hilbert schemes corresponding to smooth Fano threefolds, we will encounter three non-smoothing components shown in Table 2. General elements of these components are certain Gorenstein trigonal Fano threefolds, classified in [PCS05]. We give a more precise description of these components in §3.3.

Let $\mathfrak{tor}_{d}$ be the set of all degree $d$ toric Fano threefolds with at most Gorenstein singularities. This set can be understood explicitly. Indeed, Gorenstein toric Fano varieties correspond to reflexive polytopes, see e.g. [CLS11, Theorem 8.3.4]. The set of all three-dimensional reflexive polytopes has been classified by Kreuzer and Skarke in [KS98]; the corresponding data may be found in the Graded Ring Database [BK]. As our main result, we determine exactly which components of $\mathcal{H}_{d}$ contain the points corresponding to elements of $\mathfrak{tor}_{d}$. This can be summarized as follows:

We now outline the proof strategy:

1. (i)

   For each $d\leq 12$, we identify a Stanley-Reisner scheme $S$ such that $[S]\in\mathcal{H}_{d}$ is a smooth point lying on the component corresponding to $V_{d}$, cf. Theorem 4.1.

2. (ii)

   If $X$ is any Gorenstein toric Fano threefold of degree $4$, $6$, or $8$, then its moment polytope admits a certain “good” unimodular triangulation, cf. Proposition 4.2. This implies that $X$ has an embedded degeneration to the Stanley-Reisner scheme $S$ of (i). Hence, $[X]$ is a smooth point on the component corresponding to $V_{d}$. For many Gorenstein toric Fano threefold of degrees $10$ and $12$, a similar argument applies.

3. (iii)

   For $X$ one of the remaining degree $10$ and $12$ threefolds, we follow a general strategy described in §2.3. Using computer-assisted methods, we determine the number and dimensions of the tangent cone of $\mathcal{H}_{d}$ at $[X]$. These components are then matched with irreducible components of $\mathcal{H}_{d}$. There are three key ingredients:

   • •

     Comparison theorems allowing for effective computation of tangent and obstruction spaces of the Hilbert functor, see §2.1;

   • •

     An understanding of the deformation theory of subvarieties of rational normal scrolls, see §2.2;

   • •

     A description of the universal family of $\mathcal{H}_{12}$ near a special point lying in the intersection of $B_{97}$, $B_{98}$, and $B_{99}$, see §6.

   The degree $10$ and $12$ cases are dealt with respectively in Theorems 5.1 and 7.1.

To the best of our knowledge, our strategy of studying the local structure of Hilbert schemes is new. We believe that these techniques will yield success in studying Hilbert schemes for Gorenstein toric Fano threefolds of higher degrees as well, although the increase in embedding dimension will lead to increased computational difficulties. We use a number of computer programs to carry out our calculations: Macaulay2 [GS], Versal Deformations [Ilt12], TOPCOM [Ram02], and 4ti2 [tt]. Supplementary material containing many of the computer calculations is available online [CI14].

A consequence of our main result is a classification of embedded degenerations of smooth Fano threefolds to Gorenstein toric Fano threefolds of degree at most twelve. Indeed, let $V$ be a general smooth Fano threefold of degree $d\leq 12$ with very ample anticanonical divisor, and $X\in\mathfrak{tor}_{d}$. Then $V$ has an embedded degeneration to $X$ if and only if $[X]$ lies on the same irreducible component of $\mathcal{H}_{d}$ as $[V]$.

Toric degenerations are connected to mirror symmetry through the ansatz of extremal Laurent polynomials, see [Prz09]. The quantum cohomology of a smooth Fano variety $V$ of dimension $n$ is conjecturally related to the Picard-Fuchs operator of a pencil $f\colon Y\to\mathbb{C}$ called a (weak) Landau–Ginzburg model for $V$. The extremal Laurent polynomial ansatz conjectures that one should be able to take $Y=(\mathbb{C}^{*})^{n}$, that is, $f$ is a Laurent polynomial. Furthermore, denoting the Newton polytope of $f$ by $\Delta_{f}$, it is expected that if $f$ gives a Landau–Ginzburg model for $V$, then $V$ degenerates to the toric variety whose moment polytope is dual to $\Delta_{f}^{*}$. Conversely, for any Fano toric variety $X$ with mild singularities smoothing to $V$, one expects to be able to find a Landau–Ginzburg model for $V$ in the form of a Laurent polynomial $f$ with $\Delta_{f}$ dual to the moment polytope of $X$. T. Coates, A. Corti, S. Galkin, V. Golyshev, and A. Kasprzyk outline a program using these ideas to classify smooth Fano varieties in [CCG⁺12].

In [Prz09], V. Przyjalkowski showed that for every smooth Fano threefold $V$ of Picard rank one, there is in fact a Laurent polynomial giving a weak Landau–Ginzburg model for $V$. Furthermore, in [ILP13], V. Przyjalkowski, J. Lewis, and the second author of the present paper showed that these Laurent polynomials are related to toric degenerations in the above sense. T. Coates, A. Corti, S. Galkin, and A. Kasprzyk have extended Przyjalkowski’s result to Fano threefolds of higher Picard rank in [CCGK13]. Matching up our toric degenerations with the extremal Laurent polynomials they have found would provide evidence that the existence of an extremal Laurent polynomial is coupled to the existence of a toric degeneration.

Motivated by similar considerations, S. Galkin classified all degenerations of smooth Fano threefolds to Fano toric varieties with at most terminal Gorenstein singularities in [Gal07]. This situation is however significantly different than the present one. Indeed, any Fano threefold with at most terminal Gorenstein singularities has a unique smoothing. This is no longer true if we relax the condition that the singularities be terminal; smoothings need not exist, and if they do, need not be unique.

We thank Cinzia Casagrande, Sergei Galkin, Jan Stevens, and the anonymous referee for helpful comments.

Let $S=k[x_{0},\dots,x_{n}]$, $A=S/I$ be a graded ring and $X=\operatorname{Proj}A\subseteq\mathbb{P}^{n}$. We may consider two deformation functors for $X$: the deformation functor $\operatorname{Def}_{X}$ of isomorphism classes of deformations of $X$ as a scheme, and the local Hilbert functor $H_{X}$ of embedded deformations of $X$ in $\mathbb{P}^{n}$, see [Ser06] for details. The former has a formal semiuniversal element and the latter a formal universal element. There is a natural forgetful map $H_{X}\to\operatorname{Def}_{X}$. Let $T^{1}_{X}$ and $T^{2}_{X}$ be the tangent space and obstruction space for $\operatorname{Def}_{X}$ and $T^{i}_{X/\mathbb{P}^{n}}$, $i=1,2$, the same for $H_{X}$.

Assume now that $A$ is Cohen-Macaulay of Krull dimension $4$ which is the case for all schemes in this paper. We may use the comparison theorems of Kleppe to relate the $T^{i}_{X/\mathbb{P}^{n}}$ and $T^{i}_{X}$ to the degree 0 part of cotangent modules of the algebra $A$. This has a large computational benefit. For $H_{X}$, [Kle79, Theorem 3.6] applied to our situation yields

For $\operatorname{Def}_{X}$, [Kle79, Theorem 3.9] applied to our situation yields

where the latter sequence is exact. The cohomology $H^{3}(X,\mathcal{O}_{X})$ does not appear in the statement in [Kle79], but a careful reading of the proof shows the existence of the sequence.

Thus if $H^{3}(X,\mathcal{O}_{X})=0$, as will be the case for the Fano schemes in this paper, also ${(T^{2}_{A})}_{0}\simeq T^{2}_{X}$. The Zariski-Jacobi sequence for $k\to S\to A$ reads

and since $S$ is regular $T^{i}_{S}(A)=0$ for $i\geq 1$. This gives the above written isomorphisms of obstruction spaces $T^{2}_{A/S}\simeq T^{2}_{A}$ but also a surjection $T^{1}_{A/S}\to T^{1}_{A}$. By the above this means the forgetful map $H_{X}\to\operatorname{Def}_{X}$ is surjective on tangent spaces and injective on obstruction spaces, so it is smooth.

The outcome of all this is that we may do versal deformation and local Hilbert scheme computations using the vector spaces $(T^{1}_{A})_{0}$, $(T^{1}_{A/S})_{0}$ and $(T^{2}_{A})_{0}$. Moreover, by smoothness of the forgetful map, the equations for the Hilbert scheme locally at $X$, in particular the component structure, will be obstruction equations for $\operatorname{Def}_{X}$ which involves much fewer parameters.

When studying deformations of a scheme $X$, it is often useful to have a systematic way for writing down the equations in $X$. For subvarieties of rational normal scrolls, this is found in the method of rolling factors introduced by Duncan Dicks, see e.g. [Dic88] and [Ste01, §1]. Since many of the Fano varieties we consider are subvarieties of scrolls, we summarize this method in the following.

Let $d_{0}\geq d_{1}\geq\ldots\geq d_{k}$ be non-negative integers and $d=\sum d_{i}$. Let $S$ be the image of

under the map defined by the twisting bundle $\mathcal{O}(1)$. Then $\widetilde{S}$ is a $\mathbb{P}^{k}$ bundle over $\mathbb{P}^{1}$, and $S\subset\mathbb{P}^{d+k}$ is cut out by the $2\times 2$ minors of

where $l$ is the largest integer such that $d_{j}\neq 0$. We call $S$ a scroll of type $(d_{0},d_{1},\ldots,d_{k})$. Note that $\widetilde{S}=S$ if and only if $d_{k}\neq 0$.

Let $f_{0}$ be a homogeneous polynomial in the variables $x_{j}^{(i)}$, $0\leq i\leq k$, $0\leq j\leq d_{i}$ and suppose that every monomial in $f_{0}$ contains a factor from the top row of $M$. Then for every term in $f_{0}$, we may replace some $x_{j}^{(i)}$ by $x_{j+1}^{(i)}$ to obtain a new polynomial $f_{1}$. This process is called rolling factors. Different choices of the factors might lead to different polynomials $f_{1}$, but any difference is contained in the ideal generated by the $2\times 2$ minors of $M$.

Let $f_{0}$ be a homogeneous polynomial as above of degree $e$, and suppose that we can subsequently roll factors $m$ times to get polynomials $f_{0},\ldots,f_{m}$. Then the subvariety $X$ of $S$ cut out by the polynomials $f_{0},\ldots,f_{m}$ is a divisor of type $eD-mF$, where $D$ is the hyperplane class and $F$ is the image of the fiber class of $\widetilde{S}$. Furthermore, any such subvariety may be described in this matter.

Using this format for writing the equations of $X$, many of its deformations may be readily described, see [Ste01]. Arbitrary perturbations of $f_{0}$ which still may be rolled $m$ times describe deformations of $X$ within its divisor class on $S$. Such deformations are called pure rolling factor deformations. Perturbations of the entries of $M$ together with perturbations of the $f_{i}$ may give deformations of $X$ sitting on a deformed scroll. These deformations are called scrollar. In general, there are also non-scrollar deformations of $X$ which may not be described in either manner. For an illustration of all three types of deformations, see the example in §5.

Let $X$ be a subscheme of $\mathbb{P}^{n}$. We would like to identify which components of the corresponding Hilbert scheme $\mathcal{H}$ the point $[X]$ lies on. In the following, we outline a general strategy for doing this.

1. (i)

   Use obstruction calculus and the package Versal Deformations to find the lowest order terms of obstruction equations for $X$. This will be feasible in the cases of interest to us due to the comparison theorems mentioned in §2.1. Let $Z$ denote the subscheme of the affine space $\operatorname{Spec}S^{\bullet}H^{0}(X,\mathcal{N}_{X/\mathbb{P}^{n}})$ cut out by these equations; the tangent cone of $\mathcal{H}$ at $[X]$ is contained in $Z$.

2. (ii)

   Do a primary decomposition of these lowest order terms to find the irreducible decomposition $Z_{1},\ldots,Z_{k}$ of $Z$. Any component of the tangent cone $\operatorname{TC}_{[X]}\mathcal{H}$ is contained in some $Z_{i}$. Let $d_{i}$ denote the dimension of $Z_{i}$.

3. (iii)

   For each $Z_{i}\subset Z$, find a tangent vector $v\in H^{0}(X,\mathcal{N}_{X/\mathbb{P}^{n}})$ such that $v\in Z_{i}$ but $v\notin Z_{j}$ for $j\neq i$. Use Versal Deformations to lift the first order deformation given by $v$ to higher order to get a one-parameter deformation $\pi:\mathcal{X}\to\mathbb{A}^{1}$ of $X$. In general, this may not be possible since the process of lifting to higher order may never terminate, resulting in a family defined by a power series. In practice however, for judicious choice of $v$, we almost always get a polynomial lifting after finitely many steps.

4. (iv)

   We consider a general fiber $X^{\prime}=\mathcal{X}_{t}$, $t\neq 0$ of $\mathcal{X}$. Suppose that $h^{0}(X^{\prime},\mathcal{N}_{X^{\prime}/\mathbb{P}^{n}})=d_{i}$ and $T^{2}_{X^{\prime}/\mathbb{P}^{n}}=0$. Then $[X^{\prime}]$ lies on a component $B$ of $\mathcal{H}$ with $\dim B=d_{i}$. This implies that $Z_{i}$ is a component of $\operatorname{TC}_{[X]}\mathcal{H}$. Indeed, $[X]$ must also lie on $B$, and $\operatorname{TC}_{[X]}\mathcal{H}$ must have a component $Z_{i}^{\prime}$ of dimension $d_{i}$ which contains $v$, since $X$ deforms to $X^{\prime}$ with tangent direction $v$. Because $Z_{i}^{\prime}$ contains $v$, it must be contained in $Z_{i}$, and equality follows from the equality in dimension.

5. (v)

   Suppose that we have shown that $Z_{i}$ is a component of $\operatorname{TC}_{[X]}\mathcal{H}$ as described in step (iv). We now wish to determine for which component $B$ of $\mathcal{H}$ the tangent cone $\operatorname{TC}_{[X]}B$ contains $Z_{i}$. One approach is via deformation of the $X^{\prime}$ above: if $X^{\prime}$ deforms to some scheme $V$ for which we know $[V]$ lies on $B$, then $[X^{\prime}]$ lies on $B$ and $Z_{i}\subset\operatorname{TC}_{[X]}B$. A slightly more complicated approach is via degeneration of $X^{\prime}$: suppose that $X^{\prime}$ degenerates to a scheme $X_{0}$. If there is a component $B$ of $\mathcal{H}$ such that the degeneration direction from $X^{\prime}$ to $X_{0}$, viewed as a deformation of $X_{0}$, only lies in $\operatorname{TC}_{[X_{0}]}B$ and no other components of $\operatorname{TC}_{[X_{0}]}\mathcal{H}$, then $[X^{\prime}]$ lies on $B$ and again $Z_{i}\subset\operatorname{TC}_{[X]}B$.

Several difficulties may arise when attempting to put the above strategy into practice. For one, limits on computer memory and processor speed might make obstruction or primary decomposition calculations impossible. Secondly, it could occur that the scheme $Z$ is not equal to $\operatorname{TC}_{[X]}\mathcal{H}$; this means that there will be some $Z_{i}$ which strictly contains a component of $\operatorname{TC}_{[X]}\mathcal{H}$. Thirdly, as mentioned in step (iii), lifting of one-parameter first order deformations might not terminate.

For all the cases of present interest, these three problems almost never arise. The only such problem we will encounter is in the few cases where some $Z_{i}$ is an embedded component. In these cases we can use deformation considerations to show that $Z_{i}$ does not correspond to a smoothing component of the Hilbert scheme. This is done in the final two examples of §7.

It can also occur in step (iv) that $h^{0}(X^{\prime},\mathcal{N}_{X^{\prime}/\mathbb{P}^{n}})=d_{i}$ and $[X^{\prime}]$ is a smooth point of $\mathcal{H}$, but $T^{2}_{X^{\prime}/\mathbb{P}^{n}}\neq 0$. In such cases, an alternate strategy is needed to show that $[X^{\prime}]$ is indeed a smooth point of $\mathcal{H}$. One possible approach to deal with this problem is by using the structure of rolling factors, as we do for several cases in the proof of Theorem 5.1.

Before discussing specific components of the Hilbert schemes $\mathcal{H}_{d}$, we prove a result concerning Hilbert scheme component dimensions for Fano varieties in general. Recall from §2.1 that given a scheme $V\subset\mathbb{P}^{n}$, the dimension of the tangent space of the Hilbert scheme at the corresponding point $[V]$ is just $h^{0}(\mathcal{N}_{V/\mathbb{P}^{n}})$. For smooth Fano varieties, this can be computed as follows:

In Table 1, we list all families of smooth Fano threefolds of degree at most twelve. Degrees of general elements of these families and their topological invariants are taken from [Isk78], [Isk79], and [MM82]. Our names, referring both to the family and to general elements thereof, are non-standard. Below we calculate case by case whether the anticanonical divisor $-K_{V}$ is very ample. If so, we use Proposition 3.1 to calculate how many global sections the corresponding normal sheaf has. This gives us a list of all components of the Hilbert schemes $\mathcal{H}_{d}$ for $d\leq 12$ which correspond to smooth Fano threefolds, and the dimensions thereof. To summarize, we have the following:

The varieties $V_{4}$, $V_{6}$, and $V_{8}$ all have very ample anticanonical divisors. Indeed, in their anticanonical embeddings they are complete intersections in projective space of degrees $4$, $(2,3)$, and $(2,2,2)$. Furthermore, the varieties $V_{10}$ and $V_{12}$ have very ample anticanonical divisors, cf. [Muk04]. In its anticanonical embedding, $V_{10}$ is the intersection of the Grassmannian $G(2,5)$ in its Plücker embedding with a quadric and two hyperplanes. Likewise, $V_{12}$ is the intersection of the orthogonal Grassmannian $OG(5,10)$ in its Plücker embedding with seven hyperplanes. We now deal with the remaining cases.

In our study of $\mathcal{H}_{10}$ and $\mathcal{H}_{12}$, we will encounter three additional components which do not correspond to smooth Fano threefolds, but instead non-smoothable trigonal Fano threefolds with Gorenstein singularities, see [PCS05].

We first describe an $88$-dimensional component $B_{88}^{\dagger}$ of $\mathcal{H}_{10}$. Consider the matrix

Let $g_{0},g_{1},g_{2}$ be general quadrics in $x_{i},y_{j},z_{1},z_{2}$, and let $f_{0},f_{1},f_{2}$ be the cubics defined by

Note that $f_{1}$ and $f_{2}$ have been constructed from $f_{0}$ in a manner similar to rolling factors.

Let $I$ be the ideal generated by the $2\times 2$ minors of $M$ and $f_{0},f_{1},f_{2}$. This cuts out a singular degree $10$ Fano variety $V\subset\mathbb{P}^{7}$ corresponding to a point $[V]\in\mathcal{H}_{10}$. Indeed, this is the case $T_{3}$ of [PCS05]. Using Macaulay2, we compute that $h^{0}(V,\mathcal{N})=88$, and that all deformations of $V$ come from perturbing the quadrics $g_{0},g_{1},g_{2}$ and are unobstructed. Thus, $[V]$ is a smooth point on an $88$-dimensional component $B_{88}^{\dagger}$ of $\mathcal{H}_{10}$.

The remaining two non-smoothing components may be nicely described using rolling factors. The Hilbert scheme $\mathcal{H}_{10}$ has an additional $84$-dimensional component $B_{84}^{\dagger}$ which is not the component $B_{84}$. Consider the matrix

With additional variables $z_{1},z_{2}$, its maximal minors define a scroll of type $(2,2,0,0)$. Let $f_{0}$ be a general cubic which can be rolled $2$ times to $f_{1}$ and $f_{2}$. The ideal generated by the minors of $M$ together with these three cubics cuts out out a singular degree $10$ Fano variety $V\subset\mathbb{P}^{7}$ corresponding to a point $[V]\in\mathcal{H}_{10}$. Indeed, this is the case $T_{9}$ of [PCS05].

Using Macaulay2, we compute that $h^{0}(V,\mathcal{N})=84$, and that all deformations of $V$ are of pure rolling factor type. Thus, $[V]$ is a smooth point on a $84$-dimensional component of $\mathcal{H}_{10}$, and $V$ cannot be smoothed. It follows that the component $B_{84}^{\dagger}$ of $\mathcal{H}_{10}$ upon which $[V]$ lies is not $B_{84}$.

The Hilbert scheme $\mathcal{H}_{12}$ has an additional $99$-dimensional component $B_{99}^{\dagger}$ which is not the component $B_{99}$. Consider the matrix

With additional variables $z_{1},z_{2}$, its maximal minors define a scroll of type $(4,1,0,0)$. Let $f_{0}$ be a general cubic which can be rolled $3$ times to $f_{1},f_{2}$, and $f_{3}$. The ideal generated by the minors of $M$ together with these four cubics cuts out out a singular degree $12$ Fano variety $V\subset\mathbb{P}^{8}$ corresponding to a point $[V]\in\mathcal{H}_{12}$. Indeed, this is the case $T_{25}$ of [PCS05].

Using Macaulay2, we compute that $h^{0}(V,\mathcal{N})=99$, and that the obstruction space $T_{V}^{2}$ vanishes. Thus, $[V]$ is a smooth point on a $99$-dimensional component of $\mathcal{H}_{12}$. All deformations of $V$ are of pure rolling factor type, so $V$ cannot be smoothed. Thus, the component $B_{99}^{\dagger}$ of $\mathcal{H}_{12}$ upon which $[V]$ lies is not $B_{99}$.

Let $[n]$ be the set $\{0,\ldots,n\}$ and $\Delta_{n}$ be the full simplex $2^{[n]}$. An abstract simplicial complex is any subset $\mathcal{K}\subset\Delta_{n}$ such that if $f\in\mathcal{K}$ and $g\subset f$, then $g\in\mathcal{K}$. Elements $f\in\mathcal{K}$ are called faces; the dimension of a face $f$ is $\dim f:=\#f-1$. Zero-dimensional faces are called vertices; one-dimensional faces are called edges. The valency of a vertex is the number of edges containing it. Two simplicial complexes are isomorphic if there is a bijection of the vertices inducing a bijection of all faces. We will not differentiate between isomorphic complexes.

Given two simplicial complexes $\mathcal{K}$ and $\mathcal{L}$, their join is the simplicial complex

where $f\vee g$ denotes the disjoint union of $f$ and $g$. To any simplicial complex $\mathcal{K}\subset\Delta_{n}$, we associate a square-free monomial ideal $I_{\mathcal{K}}\subset\mathbb{C}[x_{0},\ldots,x_{n}]$

where for $p\in\Delta_{n}$, $x_{p}:=\prod_{i\in p}x_{i}$. This gives rise to the Stanley-Reisner ring $A_{\mathcal{K}}:=\mathbb{C}[x_{0},\ldots,x_{n}]/I_{\mathcal{K}}$ and a corresponding projective scheme $\mathbb{P}(\mathcal{K}):=\operatorname{Proj}A_{\mathcal{K}}$ which we call a Stanley-Reisner scheme. Certain properties of the scheme $\mathbb{P}(\mathcal{K})$ are reflected in the combinatorics of the complex $\mathcal{K}$. For example, each face $f\in\mathcal{K}$ corresponds to some $\mathbb{P}^{\dim f}\subset\mathbb{P}(\mathcal{K})$ and the intersection relations among these projective spaces are identical to those of the faces of $\mathcal{K}$. In particular, maximal faces of $\mathcal{K}$ correspond to the irreducible components of $X$.

In this paper, we will only consider Stanley-Reisner schemes of the form $\mathbb{P}(\mathcal{K}*\Delta_{0})$, where $\mathcal{K}$ is topologically the triangulation of a two-sphere. Such schemes are Gorenstein Fano threefolds, and are embedded via the anticanonical divisor, see [CI12, Proposition 2.1].

We recall the correspondence between unimodular triangulations and degenerations of toric varieties. Consider some lattice $M$ and some lattice polytope $\nabla\subset M_{\mathbb{Q}}$ in the associated $\mathbb{Q}$-vector space. By $\mathbb{P}(\nabla)$ we denote the toric variety

where $S_{\nabla}$ is the semigroup in $M\times\mathbb{Z}$ generated by the elements $(u,1)$, $u\in\nabla\cap M$. By Theorem 8.3 and Corollary 8.9 of [Stu96], square-free initial ideals of the toric ideal of $\mathbb{P}(\nabla)$ are exactly the Stanley-Reisner ideals of unimodular regular triangulations of $\nabla$, see loc. cit. for definitions.

We now describe the triangulated two-spheres we will need. First of all, let $T_{4}=\partial\Delta_{3}$, $T_{5}=(\partial\Delta_{2})*(\partial\Delta_{1})$, and for $6\leq i\leq 10$, let $T_{i}$ be the unique triangulation of the sphere with $i$ vertices having valencies four and five.¹¹1The $T_{i}$ arise naturally as the boundary complexes of the convex deltahedra (excluding the icosahedron). For concrete realizations of these triangulations, see [CI12, Figure 1]. The corresponding Stanley-Reisner schemes $\mathbb{P}(T_{i}*\Delta_{0})$ satisfy some nice properties:

This theorem alone allows us to determine the position of toric Fano threefolds of degree $d$ with at most Gorenstein singularities in $\mathcal{H}_{d}$ for $2<d<10$: