5 10 15 20
1 10 100 1000
Figure 3.6: Histogram of the number of polytopes that have a given number of nef partitions.
There are 3090 reflexive three-dimensional polytopes that do not admit a nef partition. The reflexive polytope with PALP id 214 has the most nef partitions, namely 21.
P1× F1 F2 F3 F4 F5 F6 F7 F8
PALP id (4,2) (30,1) (29,3) (17,1) (84,8) (61,2) (218,0) (149,3) P1× F9 F10 F11 F12 F13 F14 F15 F16
PALP id (194,5) (113,0) (283,0) (356,3) (453,0) (505,0) (509,0) (768,1) Table 3.4: The PALP ids for the 16 nef partitions that are direct products inside the spaces
P1×Fi, where the Fi are the reflexive polygons defined infigure 3.7.
have created a website at
http://wwwth.mpp.mpg.de/members/jkeitel/Weierstrass/ (3.5.2)
containing a database of Weierstrass forms. In subsection 3.8.2we explain in detail how to extract the Weierstrass forms and other relevant information from the database.
66 CHAPTER 3. FIBER CURVES OF GENUS ONE
3.5.1 Exceptions in Codimension Two
It turns out that there are only two three-dimensional nef partitions (out of 3134) for which the algorithm ofsection 3.4 fails, that is, there is no line bundle on the ambient toric variety such that
• The degree deg(L|Y)≤4, and
• All required5 sections for finding the Weierstrass form are restrictions of sections from the ambient space.
The first exception is just P1 ×P2 with the nef partition D1 = O(2,1) and D2 = O(0,2).
Again using [x0 : x1] ∈ P1 and [y0 : y1 : y2] ∈ P2 as homogeneous coordinates, the two defining polynomials are
p1 =
2
X
i=0
(a00ix20+a01ix0x1+a11ix21)yi,
p2 =
2
X
i,j=0
bijyiyj =
y0 y1 y2
b00 b10 b20
b01 b11 b21
b02 b12 b22
y0
y1
y2
.
(3.5.3)
Projection onto theP1 factor defines a mapY =V(hp1, p2i)→P1. Its pre-image consists of two points: For fixed [x0 :x1]∈P1, the first equationp1 is a line and the second equationp2 is a conic in P2, which necessarily intersect in two points. These two points can degenerate to a single point with multiplicity two, and they must do so at precisely four pre-images because a torus is the double cover of P1 branched at four branch points. In other words, the discriminant δP1 of the double cover Y → P1 is a quartic in the variables x0, x1 with coefficients involvinga’s andb’s but noy’s.
The form of the discriminant is constrained by symmetry; SL(2,C) ×SL(3,C) acts naturally on the ambient space. The complete intersection Y is not invariant under this symmetry, but its Weierstrass form must be. More formally, we can combine the action on the homogeneous coordinates with an action on the coefficients such that the combined action does not change the equationsp1,p2. For example, theM3∈SL(3,C)-part of the action is
y0
y1 y2
7→M3
y0
y1 y2
,
aij0
aij1 aij2
7→M3−1
aij0
aij1 aij2
, (bij)7→(M3−1)T(bij)M3−1. (3.5.4) Acovariantis a polynomial that does not transform under the combined group action, obvious examples arep1 and p2. An invariant is a covariant that, furthermore, does not depend on
5For degree-one, we require the sections ofL,L2,L3, andL6. For degree-two, we requireL,L2, andL4. For degree-three, we requireLandL3. For degree-four, we requireLandL2.
3.6. NON-TORIC NON-ABELIAN GAUGE GROUPS 67 the homogeneous coordinates, for example det(bij). The discriminantδ1 that we are looking for must be a covariant of bi-degree (4,0) in [x0 :x1] and [y0:y1 :y2].
The tersest way to characterize δ1 completely is as the Θ0-invariant [138, 141] of the system of two conics (p21, p2). That is, ignore the action on theP1 factor for the moment and consider p21 and p2 as two quadratics in [y0 : y1 :y2]. The determinant ∆ of the coefficient matrix of a quadratic is clearly an invariant of the action onP2, hence so is every-coefficient in the formal expansion6
∆(p21+p2) = ∆(p21) +Θ(p21, p2) +2Θ0(p21, p2) +3∆(p2) (3.5.5) We note thatδ1(x0, x1) = Θ0(p21, p2) is quartic in x0 andx1, quadratic in the coefficientsaijk and quadratic in the coefficientsbij. Finally, the equation of a double cover branched at the zeroes ofδ1 is
Y2 =δ1(x0, x1), (3.5.6)
for which we already know how to write the Weierstrass form [139, 142], as it is simply a genus-one curve insideP112.
It remains to consider the second exceptional case. Geometrically, it is the product P1×dP1, that is, a simple blowup7 of the first case along a curve P1 × {pt.}. Moreover, the two divisors defining the nef partition are just the pull-backs of the two divisors of the first case. In terms of toric geometry, this means that the dual polytope∇contains the dual polytope ofP1×P2. Dually, the polytope ∆ is contained in the polytope ofP1×P2. Hence the formula for bringing the complete intersection into Weierstrass form is simply a specialization of the formula from the first case where some coefficients are set to zero.
3.6 Non-Toric Non-Abelian Gauge Groups
While studying the embedding genus-one curves with two points insection 3.2, we noted that these are mapped to singular curves in P112. The singularity could be resolved by blowing upP112 and the exceptional divisor introduced in the blow-up provided one of the homology classes of the sections.
In our approach, we generally take the reverse route: Starting with a smooth genus-one curve embedded inside a toric ambient spaceX with a givenh1,1(X), we map the curve into its Weierstrass form insideP231. The ambient spaces we use tend to have a richer homology than P231 and this process is generally a blow-down eliminating h1,1(X)−1 variables and producing a singular Weierstrass model. Generally, the blow-up divisors are of two different types on the resolved side:
6The invariants ∆(p21) and Θ(p21, p2) vanish becausep21is a degenerate conic.
7We use the notation whereP2=dP0.
68 CHAPTER 3. FIBER CURVES OF GENUS ONE
• They can resolve singularities occurring at the collision of two or more sections and provide the homology class of a multisection.
• They can resolve non-Abelian singularities, i.e. those along which the discriminant van-ishes at least quadratically. We call these non-toric non-Abelian singularities.
This section is dedicated to the study of the latter type and we proceed by examining the example given inEquation 3.5.3 further.
In order to study the non-Abelian singularities of a Weierstrass model defined by the triple (f, g,∆), one must attempt to factor f, g, and ∆. If they contain non-trivial factors along which ∆ vanishes at least to second order, then the fibration has a non-Abelian singularity along the base divisor defined by the vanishing of this factor. In the case ofEquation 3.5.3, f, g and ∆ are unfortunately too long to be displayed here and we refer to the database for the full expressions. While f and g do not have any non-trivial factors, the discriminant ∆ can be decomposed into ∆ =σ·∆0, where
σ= (b(12)b(02)b(01)−b00b2(12)−b11b2(02)−b22b2(01)+ 4b22b11b00)2 (3.6.1) b(ij)≡bij+bji, and ∆0 the remaining linear factor. Correspondingly, there is ansu(2) singu-larity along the locus Σ :σ= 0. Note furthermore thath1,1(P1×P2) = 2 and therefore there exists one more independent divisor on the resolved side than in the blown-down Weierstrass model. This additional divisor serves as the Cartan divisor of SU(2) and is a P1 fibration over the base locus Σ. In the set-ups relevant to us, the bij are sections of line bundles on the base manifold. Depending on the details of the full fibration, it is possible that σ = 0 does not have any solutions, as would for instance be the case when the polynomialσ is just a constant. Obviously, if that happens, then the singularity is not realized. Furthermore, in these cases the additional divisor class provided by the blow-up divisor will become trivial upon restriction to the complete intersection defining the genus-one curve inside X. Tori-cally, the ray corresponding to the blow-up divisor will then lie inside a facet of the reflexive polytope specifying the ambient space of the Calabi-Yau manifold. Whether or not this hap-pens depends of course on the full reflexive polytope and not only the reflexive subpolytope corresponding to the ambient fiber space.
In order to find all possible singularities that a completely generic8 fibration with fiber inside a given ambient space has, we fully resolve the fan of the toric ambient space. That is, we use every non-zero interior point of the reflexive polytope as a ray. The irreducible factors of ∆ occurring at least quadratically then constitute the set of generic non-toric non-Abelian singularities. We call these singularities non-toric, because they cause the genus-one fiber to split into multipleP1s while the toric ambient fiber space remains irreducible. This is in
8Here our notion of genericity equivalent to demanding that all factors of ∆ define divisors that are realized in the base manifold. The analogous requirement on the reflexive polytope defining the full Calabi-Yau manifold is that none of its non-zero integral points are interior points of a facet.
3.7. SECTIONS OF ELLIPTIC FIBRATIONS 69