Some examples of non-smoothable Gorenstein Fano toric threefolds
We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show examples of singular Gorenstein Fano toric threefolds which have compound Du Val, hence smoothable, singularities but are not smoothable.
The geometry of Mirror Symmetry for Fano varieties [mirror_symmetry_and_fano_manifolds] is based on the phenomenon of toric degeneration. From this point of view, it is crucial to study deformations and smoothings of Fano toric varieties. In this note we will restrict to the case of Fano toric threefolds with Gorenstein singularities; such varieties are in one-to-one correspondence with the 4319 reflexive polytopes of dimension 3, which were classified by Kreuzer and Skarke [kreuzer_skarke_reflexive_3topes].
Fix such a polytope and denote by the corresponding Fano toric variety, i.e. the toric variety associated to the spanning fan of . The singularities of are detected by the shape of the facets of . Here we will ignore the problem of understanding which singularities are smoothable. Instead, we will present a local-to-global obstruction to the smoothability of . In other words, we will show examples where there exists an open non-affine subscheme such that is singular, has smoothable singularities, and is not smoothable (and consequently is not smoothable). These examples are constructed by means of the following combinatorial criterion — the relevant definitions are given in §3.
Let be a reflexive polytope of dimension and let be the Fano toric threefold associated to the spanning fan of . If, for some integer , the polytope has “two adjacent almost-flat -triangles” as facets, then is not smoothable.
There are 273 reflexive polytopes of dimension 3 which satisfy the condition of Theorem 1.1. The complete list is given in Remark 3.8. Therefore, there are at least 273 non-smoothable Gorenstein toric threefolds.
The following example, which satisfies Theorem 1.1, exhibits a Gorenstein Fano toric threefold which is non-smoothable and has only singularities. This variety refutes a conjecture made by Prokhorov [prokhorov_degree_Fano_threefolds, Conjecture 1.9], according to which all Fano threefolds with only compound Du Val singularities are smoothable. This conjecture was motivated by Namikawa’s result [namikawa_fano] on the smoothability of Fano threefolds with Gorenstein terminal singularities.
In the lattice consider the reflexive polytope that is the convex hull of the columns of the matrix
Let be the Fano toric threefold associated to the spanning fan of . One can show that the singular locus of is a curve isomorphic to and that the singularities along are transverse . Using the techniques of §3, it is possible to prove that the sheaf is isomorphic to . This implies that
and, by Lemma 2.5, that every infinitesimal deformation of is locally trivial. Therefore is not smoothable.
Idea of the proof
If a reflexive polytope of dimension has two adjacent -triangles as facets, then there is an open non-affine toric subscheme of such that the singular locus of is isomorphic to and the singularities are transverse . Here denotes the affine toric surface . More precisely, is an -bundle over (see Definition 2.1), i.e. there exists a map such that, Zariski locally on the target, it is the trivial projection with fibre . The map may be globally non-trivial, depending on the relative position of the two adjacent -triangles. It is possible to express the sheaf , which is a vector bundle on of rank , in terms of the combinatorics of the two triangles. In particular, we get to know when this sheaf is the direct sum of negative line bundles on . This gives a combinatorial sufficient condition for not to have global sections; the condition is expressed by insisting that the two triangles almost lie on the same plane, i.e. they are “almost-flat” (see Definition 3.2). If this happens, then every infinitesimal deformation of is locally trivial and, thus, is not smoothable.
Notation and conventions
We work over , but everything will hold over a field of characteristic zero or over a perfect field of large characteristic.
If is a lattice, its dual is denoted by and the symbol denotes the duality pairing between and .
I would like to thank Alessio Corti for suggesting this problem to me and for sharing his ideas. I am grateful to Victor Przyjalkowski for bringing Prokhorov’s conjecture to my attention.
The author was funded by Tom Coates’ ERC Consolidator Grant 682603 and by Alexander Kasprzyk’s EPSRC Fellowship EP/N022513/1.
2. -bundles and their deformations
For any integer , let denote the toric surface singularity associated to the cone spanned by and inside the lattice , i.e. the affine hypersurface
The conormal sequence of the closed embedding produces a free resolution of :
where is the ideal of in . This allows us to compute
where is the closed subscheme of defined by the ideal generated by , and . Notice that is the singular locus of equipped with the schematic structure given by the second Fitting ideal of .
We want to define the notion of an -bundle and globalise this computation of the Ext group. Informally, an -bundle is a morphism which, Zariski-locally, is the projection . More precisely we have to insist that an -bundle is a closed subscheme in a split vector bundle over of rank 3.
An -bundle over a -scheme is a morphism of schemes such that there exist three line bundles , a closed embedding of -schemes
of into the total space of , and an affine open cover of satisfying the following condition: for each , there are trivializations , , and a commutative diagram of -schemes
where denotes the projection , the coordinates , and are the local sections corresponding to the trivializations above, the horizontal arrows are isomorphisms, the left vertical arrow is the restriction of the closed embedding , and the right vertical arrow is the base change of the standard embedding to .
A posteriori one can see that . This follows from the following easy fact in commutative algebra: let be a ring and be an invertible element; if the ideal of generated by coincides with the ideal generated by , then .
Let be a scheme with a line bundle . Let be the th order thickening of the zero section of the total space of , i.e. the closed subscheme of locally defined by the equation where is a nowhere vanishing local section of . Let be the projection. Then
Let be an affine open cover of which trivializes . Let be a local coordinate. Then we have the isomorphism of -schemes
Therefore is the free -module with basis , which is a local frame of .
Another way to see this is to notice that , and consequently , where is the ideal made up of elements of degree greater than . ∎
Let be a -scheme and be an -bundle, with as in Definition 2.1. Then there is an isomorphism of -modules
Assume we are in the setting of Definition 2.1, with projections and , closed embedding , and a trivialising affine open cover of with local sections .
We consider the conormal sequence of :
where is the ideal sheaf of the closed embedding . We restrict this sequence to and we get the conormal sequence of :
Since is the vector bundle whose sheaf of sections is , we have that . Therefore .
One can check that . On the intersection we have the equalities , , and , where are invertible functions such that (by Remark 2.2). Then the restriction of the map
in (2) to produces the following commutative diagram.
Therefore the sequence (2) becomes
which gives a locally free resolution of . Hence
where is the closed subscheme locally defined by . Denote with the projection. It is clear that is the th order thickening of the zero section in the total space over . By Lemma 2.3 we have
This concludes the proof of Proposition 2.4. ∎
The following lemma is well known in deformation theory.
Let be a reduced -scheme. Assume that is a local complete intersection morphism and that .
Then all infinitesimal deformations of are locally trivial. In particular, if is not smooth, then is not smoothable.
Let be the functor of infinitesimal deformations of , i.e. the covariant functor from the category of local finite -algebras to the category of sets which maps to the set of isomorphism classes of deformations of over and acts on arrows by base change. Consider the subfunctor given by the locally trivial deformations. We refer the reader to [sernesi_deformation, §2.4] for details. We want to show that the natural transformation is surjective; it is enough to show that it is smooth; hence it suffices to prove that it induces a surjection on tangent spaces and an injection on obstruction spaces (for example see [manetti_seattle, Remark 4.12]).
Let be the sheaf of derivations on . By [sernesi_deformation, Theorem 2.4.1] the tangent space of is and the tangent space of is . By [sernesi_deformation, Proposition 2.4.6], is an obstruction space for . By [sernesi_deformation, Proposition 2.4.8] or [vistoli_deformation_lci, Theorem 4.4], is an obstruction space for .
The local-to-global spectral sequence for Ext gives the following five term exact sequence
With the identifications above, the vanishing of implies that induces an isomorphism on tangent spaces and an injection on obstruction spaces. ∎
Let be a smooth -scheme and be an -bundle, with as in Definition 2.1. Assume for all .
Then all infinitesimal deformations of are locally trivial. In particular, is not smoothable.
As is a Zariski-locally trivial fibration, the sequence of Kähler differentials of is left exact and locally split:
This implies that the dual sequence, i.e. the one obtained by applying , is exact. Therefore we have an exact sequence of -modules
But the last sheaf is zero because is smooth over . Therefore we have an isomorphism of -modules between and . By Proposition 2.4 we deduce that
Conclude with Lemma 2.5. ∎
3. Toric -bundles over
Fix an integer and a 3-dimensional lattice . An -triangle in is a lattice triangle such that:
there are no lattice points in the relative interior of ;
the edges of have lattice lengths , , and ;
is contained in a plane which has height with respect to the origin, i.e. there exists a linear form such that is contained in the affine plane , where is the duality pairing between and .
If is an -triangle in the 3-dimensional lattice , consider the cone spanned by the vertices of . Then the affine toric variety associated to the cone , namely , is isomorphic to . This is a singularity.
Fix an integer and a 3-dimensional lattice . Two adjacent -triangles in are two -triangles and in such that:
is the edge of length for both and ;
and lie in the two different half-spaces of defined by the plane .
We say that and are almost-flat if , where is the vertex of the triangle not in the segment and is the linear form such that is contained in the plane .
Notice that the condition of almost-flatness is symmetric between and because .
Let be a reflexive polytope in the lattice of rank and let and be two adjacent -triangles which are facets of . The convexity of implies .
Consider the dual polytope
The dual face of (resp. ) is the vertex (resp. ) of . The dual face of the edge is the edge of . The segment has lattice length equal to .
Let and be two adjacent -triangles in a 3-dimensional lattice . We denote by and the vertices of the segment . Let (resp. ) be the vertex of (resp. ) which does not lie on (see Figure 2). Let be the toric variety associated to the fan in generated by and . The projection induces a toric morphism .
Let and be two adjacent -triangles in a 3-dimensional lattice . Then the toric morphism , constructed in Setup 3.4, is an -bundle. Moreover, if then all infinitesimal deformations of are locally trivial.
After a -transformation, in Setup 3.4 we may assume that and
for some .
Let be the lattice point on the segment between and which is the closest one to . The triangle with vertices is an empty triangle at height , so is a basis of . Without loss of generality we may assume that , and . Since on the edge between and there are lattice points, we have .
Assume for some . Since are the vertices of an empty triangle at height , they constitute a basis of . Therefore .
Since and have to be in the two different half-spaces in which the plane divides , we have , so . ∎
Proof of Proposition 3.5.
By Lemma 3.6, the ray map of is given by the matrix
One can see that the ideal of generated by the minors is itself and the ideal generated by the minors is , where . Let be such that and . The kernel of the ray map is generated by the primitive vector . By Bézout let be such that . The cokernel of the transpose of the ray map is the homomorphism given by the matrix
where denotes the reduction modulo . By [cls, Theorem 4.1.3], the divisor class group of is isomorphic to .
Let the group
act linearly on . By [cls, §5.1], is the quotient of with respect to this action. Let be the Cox coordinates of associated to the rays , respectively. The toric morphism is defined by
We consider the following integers
and we consider the line bundles , , and the sheaf on . Let be the total space of over . is the quotient with respect to the action of with weights . It is easy to check that the map given by is a closed embedding, locally defined by . So is an -bundle and we are in the situation of Definition 2.1.
The triangle is contained in the plane , where . Therefore . Hence, the inequality implies that is a negative line bundle on and, by Corollary 2.6, that all infinitesimal deformations of are locally trivial. ∎
Out of the 4319 reflexive Fano polytopes of dimension 3, there are 27 polytopes such that the toric variety associated to the spanning fan of has a singular locus isomorphic to and there is an open neighbourhood of the singular locus that is an -bundle over , for some . Using the criterion above we can deduce that 10 out of these 27 toric varieties have only locally trivial deformations. One such variety is exhibited in Example 1.2.
More generally, one can consider the Gorenstein Fano toric threefolds such that there exists a toric open immersion where is a toric -bundle over . If the condition of Proposition 3.5 is satisfied, then has only locally trivial deformations and consequently is not smoothable. This is the proof of Theorem 1.1.
We use the classification of reflexive 3-dimensional polytopes by Kreuzer and Skarke [kreuzer_skarke_reflexive_3topes], but we use the IDs that appear in http://www.grdb.co.uk. In this way we can prove that the following 273 toric Gorenstein Fano threefolds are not smoothable: 15, 16, 36, 41, 45, 53, 58, 59, 61, 65, 66, 102, 105, 110, 111, 112, 113, 116, 117, 124, 125, 128, 135, 141, 142, 144, 146, 147, 148, 149, 152, 162, 172, 179, 183, 189, 192, 193, 197, 230, 236, 244, 248, 261, 268, 271, 272, 277, 278, 279, 280, 281, 282, 286, 288, 290, 292, 302, 310, 324, 325, 327, 331, 332, 333, 334, 335, 337, 340, 343, 347, 349, 351, 355, 356, 358, 361, 362, 386, 399, 400, 407, 443, 445, 448, 452, 453, 456, 457, 463, 467, 487, 490, 496, 497, 499, 501, 502, 505, 507, 508, 509, 511, 512, 516, 523, 540, 545, 550, 563, 569, 577, 579, 581, 582, 583, 594, 599, 600, 601, 605, 606, 617, 629, 633, 658, 670, 671, 672, 674, 679, 682, 687, 705, 760, 764, 770, 771, 780, 781, 786, 787, 792, 797, 799, 809, 811, 812, 815, 816, 824, 859, 865, 868, 873, 875, 878, 883, 884, 889, 891, 892, 893, 894, 895, 902, 905, 929, 956, 960, 965, 987, 1003, 1004, 1006, 1011, 1021, 1038, 1045, 1051, 1156, 1160, 1168, 1175, 1177, 1199, 1203, 1209, 1216, 1217, 1225, 1232, 1234, 1251, 1252, 1253, 1255, 1256, 1260, 1262, 1265, 1275, 1286, 1287, 1293, 1300, 1305, 1308, 1324, 1327, 1351, 1371, 1383, 1398, 1533, 1545, 1550, 1551, 1554, 1561, 1579, 1589, 1613, 1614, 1615, 1620, 1637, 1638, 1656, 1665, 1666, 1671, 1686, 1690, 1693, 1697, 1711, 1747, 1748, 1760, 1763, 1989, 2000, 2001, 2027, 2045, 2051, 2052, 2068, 2071, 2072, 2076, 2084, 2096, 2098, 2102, 2379, 2380, 2385, 2403, 2405, 2423, 2424, 2425, 2427, 2738, 2777, 2778, 2792, 3047, 3057, 3063, 3064.