The Donaldson-Thomas Theory of the Banana Threefold with Section Classes

We further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied in [Bryan'19]. These are smooth proper Calabi-Yau threefolds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a"banana configuration''. In [Bryan'19] the Donaldson-Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande-Thomas theory for a rational elliptic surface and present new Gopakumar-Vafa invariants for the banana threefold.

For a non-singular Calabi-Yau threefold Y over C we let be the Hilbert scheme of one dimensional proper subschemes with fixed homology class and holomorphic Euler characteristic. We can define the (β, n) Donaldson-Thomas invariant of Y by: Behrend proved the surprising result in [Be Be] that the Donaldson-Thomas invariants invariants are actually weighted Euler characteristics of the Hilbert scheme: DT β,n (Y ) = e(Hilb β,n (Y ), ν) := k∈Z k · e(ν −1 (k)).
Here ν : Hilb β,n (Y ) → Z is a constructible function called the Behrend function and its values depend formally locally on the scheme structure of Hilb β,n (Y ). We also define the unweighted Donaldson-Thomas invariants to be: DT β,n (Y ) = e(Hilb β,n (Y )).
These are often closely related to Donaldson-Thomas invariants and their calculation provides insight to the structure of the threefold. Moreover, many important properties Donaldson-Thomas invariants such as the PT/DT correspondence and the flop formula also hold for the unweighted case [T1 T1, T2 T2].
The depth of Donaldson-Thomas theory is often not clear until one assembles the invariants into a partition function. Let {C 1 , . . . , C N } be a basis for H 2 (Y, Z), chosen so that if β ∈ H 2 (Y, Z) is effective then β = d 1 C 1 + · · · + d N C N with each d i ≥ 0. The Donaldson-Thomas partition function of Y is: We also define the analogous partition function Z for the unweighted Donaldson-Thomas invariants. On the right the two rational elliptic surfaces S 1 and S 2 are highlighted.
Remark 1.1.1. This choice of variable is not necessarily the most canonical as shown in [Br Br] where the variable p is substituted for −p. However, in this article we will be focusing on the unweighted Donaldson-Thomas invariants where this choice makes the most sense.
This partition function is very hard to compute and for proper Calabi-Yau threefolds, the only known complete examples are in computationally trivial cases. However, when we restrict our attention to subsets of H 2 (Y, Z) there are many remarkable results. Two case which we will be related to computations are the Schoen (Calabi-Yau) threefold of [S S] and the banana (Calabi-Yau) threefold of [Br Br].
We will employ computational techniques developed in [BK BK] for studying Donaldson-Thomas theory of local elliptic surfaces.
1.2. Donaldson-Thomas Theory of Banana Threefolds. The banana threefold is of primary interest to us and is defined as follows. Let π : S → P 1 be a generic rational elliptic surface with a section ζ : P 1 → S. We will take S to be P 2 blown-up at 9 points which gives rise to 9 natural choices for ζ. The associated banana threefold is the blow-up (1) where ∆ is the diagonal divisor in S × P 1 S. The surface S is smooth but the morphism π : S → P 1 is not. It is singular at 12 points of S and this gives rise to 12 conifold singularities of S × P 1 S that all lie on the divisor ∆. This makes X a conifold resolution of S × P 1 S. It is a non-singular simply connected proper Calabi-Yau threefold as shown in [Br Br,Prop. 28].
The section ζ : P 1 → S gives a section σ : P 1 → X of the natural map pr : X → P 1 . It also gives natural sections of the projections pr i : X → S which we denote by S 1 and S 2 . These are both divisors of X that are copies of the rational elliptic surface. The diagonal ∆ and anti-diagonal ∆ op of S × P 1 S are also divisors which are copies of S. The anti-diagonal intersects the diagonal in a curve on ∆ op , so it is unaffected  by the blow-up. We denote the anti-diagonal divisor in X by S op and the proper transform of the diagonal by S ∆ . The latter is a rational elliptic surface blown-up at the 12 nodal points of the fibres.
The generic fibres of the map pr : X → P 1 are Abelian surfaces of the form E × E where E = π −1 (x) is an elliptic curve that is the fibre of a point x ∈ P 1 . The projection map pr also has 12 singular fibres which are non-normal toric surfaces. They are each compactifications of C * × C * by a banana configuration and their normalisations are isomorphic to P 1 × P 1 blown up at 2 points [Br Br,Prop. 24].
Definition 1.2.1. A banana configuration is a union of three curves C 1 ∪ C 2 ∪ C 3 where C i ∼ = P 1 with N Ci/X ∼ = O(−1) ⊕ O(−1) and C 1 ∩ C 2 = C 1 ∩ C 3 = C 2 ∩ C 3 = {z 1 , z 2 } where z 1 , z 2 ∈ X are distinct points. Also, there exist formal neighbourhoods of z 1 and z 2 such that the curves C i become the coordinate axes in those coordinates. We label these curves by their intersection with the natural surfaces in X. That is C 1 is the unique banana curve that intersects S 1 at one point. Similarly, C 2 intersects S 2 and C 3 intersects S op .
The banana threefold contains 12 copies of the banana configuration. We label the individual banana curves by C (j) i (and simply C i when there is no confusion). The banana curves C 1 , C 2 , C 3 generate a sub-lattice Γ 0 ⊂ H 2 (X, Z) and we can consider the partition function restricted to these classes: In [Br Br,Thm. 4], this rank three partition function is computed to be: where d = (d 1 , d 2 , d 3 ) and the second product is over k ∈ Z unless d = (0, 0, 0) in which case k > 0. (Note the change in variables from [Br Br].) The powers c( d , k) are defined by ∞ a=−1 k∈Z c(a, k)Q a y k := k∈Z Q k 2 (−y) k Remark 1.2.2. We can pass to the unweighted Z Γ0 from the weighted partition function Z Γ0 by the change of variables Q i → −Q i and p → −p.
We can include the class of the section σ to generate a larger sub-lattice Γ ⊂ H 2 (X, Z). The partition function of this sub-lattice is currently unknown. The purpose of this article is to make progress towards understanding this partition function. We will be calculating the unweighted Donaldson-Thomas theory in the classes: β = σ + (0, d 2 , d 3 ) := σ + 0 C 1 + d 2 C 2 + d 3 C 3 , by computing the following the partition function where Z (0,•,•) is the Q 0 1 part of the unweighted version of the Γ 0 partition function (2 2) and is given by: The connected unweighted Pandharipande-Thomas version of the above formula is identified as the connected version of the Pandharipande-Thomas theory for a rational elliptic surface [BK BK,Cor. 2] in the following corollary.
Corollary B The connected unweighted Pandharipande-Thomas partition function is: Z PT,Con σ+(0,•,•) := log Z σ+(0,•,•) We will also be computing the unweighted Donaldson-Thomas theory in the classes: and the permutations involving C 1 , C 2 . So for i, j ∈ {0, 1} we define The formulas will be given in terms of the functions which are defined for g ∈ Z: Theorem C The above unweighted Donaldson Thomas functions are: (2) Z •σ+(0,1,•) = Z •σ+(1,0,•) is: The connected unweighted Pandharipande-Thomas versions of the above formula contain the same information but are given in the much more compact form. In fact we can present the same information in an even more compact form using the unweighted Gopakumar-Vafa invariants n g β via the expansion As noted before, these express the same information as the above generating functions. For β = (d 1 , d 2 , d 3 ), these invariants are given in [Br Br,§A.5]. We present the new invariants for β = bσ The unweighted Gopakumar-Vafa invariants n g β are given by: (1) If b > 1 we have n g β = 0.
(2) If b = 1 then the non-zero invariants are given in the following table: 3 appearing in the rank 3 Donaldson-Thomas partition function of [Br Br,Thm. 4]. However, there is no immediate geometric explanation for this fact.
Corollaries B B and D D will be proved in section 6.1 6.1.
1.3. Notation. The main notations for this article have been defined above in section 1.2 1.2. In particular X will always denote the banana threefold as defined in equation (1 1).
1.4. Future. The calculation here is for the unweighted Donaldson-Thomas partition function. However, the method of [BK BK] also provides a route (up to a conjecture) of computing the Donaldson-Thomas partition function. The following are needed in order to convert the given calculation: (1) A proof showing the invariance of the Behrend function under the (C * ) 2action used on the strata. (2) A computation of the dimensions of the Zariski tangent spaces for the various strata. A comparison of the unweighted and weighted partition functions of the rank 3 lattice of [Br Br] reveals the likely differences: In the variables chosen in this article one can pass from the unweighted to the weighted partition functions by the change of variables Q i → −Q i and p → −p.
Moreover, the conifold transition formula reveals further insight by comparing to the Donaldson-Thomas partition function of the Schoen variety with a single section and all fibre classes, which was shown in [OP OP] (via the reduced theory of the product of a K3 surface with an elliptic curve) to be given by the weight 10 Igusa cusp form.
As we mentioned previously the Donaldson-Thomas partition function is very hard to compute. So much so that for proper Calabi-Yau threefolds, the only known complete examples are in computationally trivial cases. This is even true conjecturally and even a conjecture for the rank 4 partition function is highly desirable. The work here shows underlying structures that a conjectured partition function must have.
2. Overview of the Computation 2.1. Overview of the Method of Calculation. We will closely follow the method of [BK BK] developed for studying the Donaldson-Thomas theory of local elliptic surfaces. However, due to some differences in geometry a more subtle approach is required in some areas. In particular, the local elliptic surfaces have a global action which reduces the calculation to considering only the so-called partition thickened curves.
Our method is based around the following continuous map: which takes a one dimensional subscheme to its 1-cycle. The fibres of this map are of particular importance and we denote them by Hilb • Cyc X, q where q ∈ Chow 1 (X). The bullet notation will be elaborated on further in this section.
Remark 2.1.1. No such morphism exists in the algebraic category. In fact we note from [K K,Thm. 6.3] that there is only a morphism from the semi-normalisation Hilb 1 (X) SN → Chow 1 (X). However, Hilb 1 (X) SN is homeomorphic to Hilb 1 (X), which gives rise to the above continuous map.
Broadly, we will be calculating the Euler characteristics e Hilb β,n (X) using the following method: (1) Push forward the calculation to an Euler characteristic on Chow 1 (X), weighted by the constructible function (Cyc * 1)(q) := e Hilb • Cyc (X, q) . This is further described in sections 2.2 2.2 and 2.3 2.3.
(2) Analyse the image of Cyc and decompose it into combinations of symmetric products where the strata are based on the types of subscheme in the fibres Hilb • Cyc (X, q). This is done in section 3 3. (3) Compute the Euler characteristic of the fibres e Hilb • Cyc (X, q) and show that they form a constructible function on the combinations of symmetric products. This is done in section 5 5.
(4) Use the following lemma to give the Euler characteristic partition function. Lemma 2.1.2. [BK BK,Lemma 32] Let Y be finite type over C and let g : Z ≥0 → Z((p)) be any function with g(0) = 1. Let G : Sym d (Y ) → Z((p)) be the constructible function defined by To compute the Euler characteristics of the fibres (Cyc * 1)(q) := e Hilb • Cyc (X, q) we use the following method made rigorous in section 4 4: Figure 5. A depiction of how the topological vertex is applied to calculate Euler characteristic of a given strata.
(1) Consider the image of the fibre under the constructible morphism denoted κ : Hilb 1 (X) → Hilb 1 (X) which takes a subscheme Z to the maximal Cohen-Macaulay subscheme Z CM ⊂ Z.
The Euler characteristic calculation of e Hilb • CM (X, q) (C * ) 2 , κ * 1 for theorems A A and C C follow similar methods but have different decompositions. The calculations are completed by considering the different types of topological vertex that occur for each fixed point in Hilb • CM (X, q) (C * ) 2 .
Since the fixed locus Hilb • CM (X, q) (C * ) 2 will be disjoint we can consider individual subschemes C ∈ Hilb • CM (X, q) (C * ) 2 and their contribution to the Euler characteristic e Hilb • CM (X, q) (C * ) 2 , κ * 1 . To compute the contribution from C we must decompose X as follows: (1) Take the complement W = X \ C.
(2) Consider, C , the set of singularities of the underlying reduced curve.
(3) Define C • = C red \ C to be its complement. The curve C will be partition thickened. So each formal neighbourhood of a point x ∈ C will give rise to a 3D partition. Similarly points on U ∈ C • will also give rise to 3D partitions and points on W will give rise to the empty partition. Using techniques from section 4.5 4.5 the Euler characteristics can then be determined.
This calculation for theorem A A is finalised in section 5.1 5.1. Generalities for the proof of theorem C C are given in section 5.2 5.2 and the individual calculations are given in sections 5.3 5.3, 5.4 5.4 and 5.5 5.5.

2.2.
Review of Euler characteristic. We begin by recalling some facts about the (topological) Euler characteristic. For a scheme Y over C we denote by e(Y ) the topological Euler characteristic in the complex analytic topology on Y . This is independent of any non-reduced structure of Y , is additive under decompositions of Y into open sets and their complements, and is multiplicative on Cartesian products. In this way we see that the Euler characteristic defines a ring homomorphism from the Grothendieck ring of varieties to the integers: If Y has a C * -action with fixed locus Y C * ⊂ Y the Euler characteristic also has the property e(Y C * ) = e(Y ).
The interaction of Euler characteristic with constructible functions and morphisms also plays a key role in this article. Recall that a function µ : T → Z is constructible if µ(T ) is finite and µ −1 (c) is the union of finitely many locally closed sets for all non-zero c ∈ µ(T ). The µ-weight Euler characteristic is defined to be e(Y, µ) = k∈Z k · e(µ −1 (k)). Note that we have e(Y ) = e(Y, 1) where 1 is the constant function.
For a scheme Z over C, a constructible morphism f : Y → Z is a finite collection of morphisms f i : Y i → Z i where Y = i Y i and Z = i Z are decompositions into locally closed subschemes. We can defined a constructible function f * µ : Z → Z by This has the important property e(Z, f * µ) = e(Y, µ). If ν : Z → Z is a constructible function, then µ · ν is a constructible function on Y × Z and e(Y × Z, µ · ν) = e(Y, µ) · e(Z, ν).
It will also be helpful to extend these definitions to the rings of formal power series in Q i and formal Laurent series in p. This will allow us to make use of lemma 2.1.2 2.1.2.
2.3. Pushing Forward to the Chow Variety. Recall that the Chow scheme Chow 1 (X) is a space parametrising the one dimensional cycles of X. We will consider the subspace of this Chow β (X) parametrising 1-cycles in the class β ∈ H 2 (X, Z). We will then define a constructible morphism ρ β : n p n Hilb β,n (X) → Chow β (X).
The strategy for calculating the partition functions is to analyse Chow β (X) and the fibres of the map ρ β . These will often involve the symmetric product, and where possible we will apply lemma 2.1.2 2.1.2.
It will be convenient to employ the following • notations for the Hilbert schemes: and for the Chow schemes: where i, j ∈ {0, 1}. Note, that here he have viewed the Hilbert and Chow schemes in the Grothendieck ring of varieties. We also extend the • notation to symmetric products in the following way: an we use the following notation for elements of the symmetric product where y i are distinct points on Y and a i ∈ Z ≥0 . We also denote a tuple of partitions α of a tuple of non-negative integers a by α a.
Using the •-notation for the maps ρ β we create the following constructible morphisms: and we also use the notation η • = η 00 • + η 01 • + η 11 • . The fibres of these morphisms will be subspaces of the Hilbert scheme parametrising one dimensional subschemes with a fixed 1-cycle. Specifically, let C ⊂ X be a one dimensional subscheme in the class β ∈ H 2 (X) with 1-cycle Cyc(C). Define Hilb n (X, Cyc(C)) ⊂ Hilb β,n (X) to be the subscheme.
The maps ρ • and η • are explicitly described in lemmas 3.5.3 3.5.3 and 3.5.1 3.5.1 respectively.
3. Parametrising Underlying 1-cycles 3.1. Related Linear Systems in Rational Elliptic Surfaces. In this section we consider some basic results about linear systems on a rational elliptic surface. Some of these result can be found in [BK BK, §A.1].
Recall our notation that π : S → P 1 is a generic rational elliptic surface with a canonical section ζ : P 1 → S. Consider the following classical results for rational elliptic surfaces from [Mi Mi,II.3]: After applying the projection formula we have the following: as well as and Lemma 3.1.1. We have the following isomorphisms: Proof. The second isomorphism is immediate from the vanishing of R i π * O S (ζ +dF ) for i > 0 (see for example [H H,III Ex. 8.1]) and H 0 (P 1 , O P 1 (d)) ∼ = 0.
To show the first isomorphism we consider the following exact sequence arising from the Leray spectral sequence: We have from (3 3) that H 1 (P 1 , π * O S (dF )) ∼ = 0 and we have the desired isomorphism after considering (4 4).
Lemma 3.1.2. Consider a fibre F of a point z ∈ P 1 by the map S → P 1 and the image of a section ζ : P 1 → S. Then there are isomorphisms of the linear systems Proof. The isomorphism |ζ + d F | S ∼ = |d z| P 1 is immediate from the vanishing of R i π * O S (ζ + dF ) for i > 0 and (3 3) (see for example [H H,III Ex. 8.1]).
We continue by showing |d F | S ∼ = |ζ + d F | S . Consider the long exact sequence arising from the divisor sequence for ζ twisted by O S (ζ + dF ): where we have applied the results from lemma 3.1.1 3.1.1. from intersection theory we have that ζ * O P 1 (ζ + dF ) ∼ = ζ * O P 1 (d − 1). So g and hence f are isomorphisms.
The isomorphism |b ζ +F | S ∼ = |z| P 1 will follow inductively from the divisor sequence for ζ on S: Intersection theory shows us that O ζ (k + 1)ζ + F is a degree −k line bundle on P 1 which shows that its 0th cohomology vanishes. Hence, we have isomorphisms: 3.2. Curve Classes and 1-cycles in the Threefold. Recall from definition 1.2.1 1.2.1 that the banana curves C i are labelled by their unique intersections with the rational elliptic surfaces S 1 , S 2 and S op . These are smooth effective divisors on X. Hence a curve C in the class (d 1 , d 2 , d 3 ) will have the following intersections with these divisors: The full lattice H 2 (X, Z) is generated by C 1 , C 2 , C 3 , σ 11 , σ 12 , . . . , σ 19 , σ 21 , . . . , σ 99 where the σ ij are the 81 canonical sections of pr : X → P 1 arising from the 9 canonical sections of π : S → P 1 . However, there are 64 relations between the σ ij 's giving the lattice rank of 20 (see [Br Br,Prop. 28 and Prop. 29]).
Proof. Any such relation must push forward to relations on S via the projections pr i : X → S i . However, S is isomorphic to P 1 blown up at 9 points. The exceptional divisors of these blow-ups correspond to the sections ζ i : P 1 → S. Hence Pic S ∼ = Pic P 2 × ζ 1 × · · · × ζ 9 ∼ = Z 10 and there are are no relations of this form.
The next lemma allows us to consider the curves in our desired classes by decomposing them.
(1) Let C be a Cohen-Macaulay curve in the class (d 1 , d 2 , d 3 ). Then the support of C is contained in fibres of the projection map pr : X → P 1 .
where C σ is a curve in the class b σ and C 0 is a curve in the class (i, j, d 3 ). The same result holds for permutations of b σ + (i, j, d 3 ).
Proof. Consider a curve in one of the given classes and it's image under the two projections pr i : X → S i . For (1) these must be in the classes |d 1 f 1 | and |d 1 f 1 |, for (2) the classes |ζ + d 1 F 1 | and |ζ + d 2 F 2 |, and for (3) the classes |if 1 | and |jf 1 |. Lemma 3.1.2 3.1.2 now shows that the curve must have the given form.
Consider a curve C with underlying 1-cycle contained in E × E, then this gives rise to a divisor D in E × E. Hence we must analyse divisors in E × E and their classes in X. The class of such a curve is determined uniquely by its intersection with the surfaces S 1 , S 2 and S op .
(1) If C is in the class (0, d 2 , d 3 ) then d 2 = d 3 and D is the pullback of a degree d 2 divisor on E via the projection to the second factor.
(2) The result in (2) is true for (d 1 , 0, d 3 ) and projection to the first factor.
Proof. If C is in the class (0, d 2 , d 3 ) then it doesn't intersect with the surface S 2 . When we restrict to E ×E this is the same condition as not intersecting with a fibre of the projection to the second factor. The only divisors that this is true for are those pulled back from E via this projection. It is clear that that the intersection with S 2 is d 2 , and that the intersection with S op is d 2 as well. Hence we have that d 2 = d 3 . The proof for part (2) is completely analogous. (c) d = 2 occurs when D is the union of a fibre from the projection to the first factor and a fibre from the projection to the second factor. (2) If j(E) = 1728 and E ∼ = C/i we have the cases (a) to (c) as well as: we have the cases (a) to (c) as well as: Proof. Denote the projection maps by p i : E × E → E and let C ⊂ X be in the class (1, 1, d) and correspond to a divisor D in E × E. Suppose D is reducible. Then from lemma 3.3.1 3.3.1 we see that D must be the union p −1 1 (x 1 ) ∪ p −1 2 (x 2 ) where x 1 , x 2 ∈ E are generic points. We also have that D is in the class (1, 1, 2).
Suppose D is irreducible. The surfaces S 1 and S 2 intersect D exactly once and their restrictions correspond the fibres of the projection maps p i : E × E → E. So the projection maps must be isomorphisms when restricted to D. Hence D is the translation of the graph of an automorphism of E.
All elliptic curves have the automorphisms x → ±x. Also we have: = 0 since one copy can be translated away from the other.
3.4. Analysis of 1-cycles in Singular Fibres of pr. We denote the fibres of the projection pr by F x := pr −1 (x). The singular fibres are all isomorphic so we denote a singular fibre by F ban and its normalisation by ν : F ban → F ban . From [Br Br,Prop. 24] we have that F ban ∼ = Bl 2 pt (P 1 × P 1 ) and if we choose the coordinates on the P 1 's so that the 0 and ∞ map to a nodal singularity, then the two points blown-up are z 1 = (0, ∞) and z 2 = (∞, 0).
be the 12 nodal fibres with the nodes.
3.4.1. Denote the divisors in F ban corresponding to the banana curve C i by C i and C i . They are identified in F ban by For i = 1, 2 we also denote C i = bl( C i ) and C i = bl( C i ) inside P 1 × P 1 . The curve classes in F ban are generated by the collection of C i and C i 's with the relations:

3.4.2.
Let f 1 and f 2 be fibres of the projections P 1 × P 1 → P 1 not equal to any C i or C i and letf 1 andf 2 be their proper transforms. Then we also have the relations: On the left is a depiction of the normalisation F ban and on the right is a depiction of P 1 × P 1 . Here bl is the map blowing up (0, ∞) and (∞, 0). On the right f 1 and f 2 are generic fibres of the projection maps P 1 × P 1 → P 1 and on the leftf 1 andf 2 are their proper transforms.
(1) C is in the class (0, 0, d 3 ) if and only if D has 1-cycle d 3 C 3 . ( whereD is the pullback of a degree a f divisor from the smooth part of N 2 such that a f + a 2 = D 2 and a f + a 3 = D 3 . Moreover, D is in the class (0, a f , a f ).
Proof. Let C ⊂ X be a curve in the class (0, d 2 , d 3 ) and correspond to a divisor D in F ban . There exists a divisor D in F ban ∼ = Bl z1,z2 (P 1 × P 1 ) with ν( D) = D.
From the discussion in 3.4.2 3.4.2 we have that bl( D) is in the class of d 2 f 2 and is hence in its corresponding linear system. So, D is the union of the the proper transform of bl( D) and curves supported at C 3 and C 3 . The result now follows.
Lemma 3.4.4. Let C ⊂ X be an irreducible curve in the class (1, 1, d) and correspond to a divisor D in F ban . Then D is the image under ν of the proper transform under bl of a smooth divisor in (1) (0, 0) and (∞, ∞) only, then d = 2.
(5) no points of P, then d = 2. Moreover, there are no smooth divisors in |f 1 + f 2 | on |P 1 × P 1 | that intersect other combinations of these points.
Proof. Let C ⊂ X be an irreducible curve in the class (1, 1, d) and correspond to a divisor D in F ban . There exists an irreducible divisor D in F ban ∼ = Bl z1,z2 (P 1 × P 1 ) with ν( D) = D. D does not contain either of the exceptional divisor C 3 and C 3 . Hence, it must be the proper transform of a curve in P 1 × P 1 .
From the discussion in 3.4.2 3.4.2 we have that bl( D) is in the class of f 1 + f 2 and is hence in its corresponding linear system. The only irreducible divisors in |f 1 + f 2 | are smooth and can only pass through the combinations of points in P that are given. We refer to the appendix 6.2.3 6.2.3 for the proof of this. The total transform in any divisor in |f 1 + f 2 | will correspond to a curve in the class C 1 + C 2 + 2C 2 . Hence the classes of the proper transforms depend the number of intersections with the set {(0, ∞), (∞, 0)}. The values are immediately calculated to be those given.
3.5. Parametrising 1-cycles. We use the notation: (1) B i = {b 1 i , . . . , b 12 i } is the set of the 12 points in S i that correspond to nodes in the fibres of the projection π : Proof. From lemma 3.2.2 3.2.2 part 2 2 it is enough to consider curves in the class (0, d 2 , d 3 ). Also from 3.2.2 3.2.2 part 1 1 we know that the curves are supported on fibres of the map pr : X → P 1 . From lemma 3.3.1 3.3.1 part 1 1 we know that the curves supported on smooth fibres of pr must be thicken fibres of the projection pr 2 : X → S. Similarly we know from lemma 3.4.3 3.4.3 part 2 2 that the curves supported on singular fibres of pr must be the union of thicken fibres of pr 2 and curves supported on the C 2 and C 3 banana curves. The result now follows.
We also use the notation: (1) N i ⊂ S i are the 12 nodal fibres of π : S i → P 1 with the nodes removed and: (3) J 0 and J 1728 to be the subsets of points x ∈ P 1 such that π −1 (x) has j-invariant 0 or 1728 respectively and J = J 0 J 1728 . (4) L to be the linear system |f 1 + f 2 | on P 1 × P 1 with the singular divisors removed where f 1 and f 2 are fibres of the two projection maps.
Remark 3.5.2. The following lemma should be parsed in the following way. For i, j ∈ {0, 1} and b, d 3 ∈ Z ≥0 , a subscheme in the class β = dσ + (i, j, d 3 ) will have 1-cycle of the following form: where D is reduced and does not contain σ or and C (i) 3 . Then D is in the class (i, j, n) for some n ∈ Z ≥0 . The Chow groups parameterise the different possible D and these possibilities depend on i and j: • If i = j = 0 then D is the empty curve. If • If i = 0 and j = 1 then D can be either a fibre of the projection pr 2 or C (i) 2 . • If i = j = i then and D then it can be combinations of fibres and banana curves. It can also be neither of these in the cases we call diagonals.
Lemma 3.5.3. In the cases β = dσ which agrees with constructible morphisms η ij • and the following decompositions of The corresponding fibres are then (η 00 (2) For i = 0 and j = 1 we have a decomposition of Chow (0,1,•) (X) with parts: . The corresponding fibres are then (η 01 (3) For i = j = 1 we have a decomposition of Chow (1,1,•) (X) with parts: where Diag • will be defined by a further decomposition. The corresponding fibres of (a)-(e) are (η 11 where L z is the proper transform of the divisor L z in P 1 × P 1 and ν is the normalisation of the kth singular fibre. Hence it is enough to parametrise the curves in the class β = (i, j, •). Also from 3.2.2 3.2.2 part 1 1 we know that the curves are supported on fibres of the map pr : X → P 1 . We must have that for some minimal reduces curve D in the class (1, 1, n) for n ≥ 0 minimal. The possible D curves are described in lemmas 3.3.1 3.  BK BK]. First we consider the following subscheme of the Hilbert scheme.
Definition 4.1.1. Let C ⊂ X be a Cohen-Macaulay subscheme of dimension 1. Consider the Hilbert scheme of subschemes Z ⊂ X of class [Z] = [C] ∈ H 2 (X) and χ(O Z ) = χ(O C ) + n for some n ∈ Z ≥0 . This contains the following closed subscheme: Hilb n (X, C) := Z ⊂ X such that C ⊂ Z and I C /I Z has finite length n .
It is convenient to replace the Hilbert scheme here with a Quot scheme. Recall the Quot scheme Quot n X (F) parametrising quotients F Q on X, where Q is zerodimensional of length n. It is related to the above Hilbert scheme in the following way.
Lemma 4.1.2. [BK BK,Lemma 5]. The following equality holds in K 0 (Var C )((p)): . We also consider the following subscheme of these Quot schemes.
Definition 4.1.3. [BK BK,Def. 12] Let F be a coherent sheaf on X, and S ⊂ X a locally closed subset. We define the locally closed subset of Quot n This allows us to decompose the Quot schemes in the following way.
Lemma 4.1.4. [BK BK,Prop. 13] Let F be a coherent sheaf on X, S ⊂ X a locally closed subset and Z ⊂ X a closed subset. Then if Z ⊂ S and and n ∈ Z ≥0 there is a geometrically bijective constructible morphism:

4.2.
An Action on the Formal Neighbourhoods. Let C ⊂ X be a one dimensional subscheme in the class β ∈ H 2 (X) with 1-cycle q = Cyc(C). We recall the our notation that Hilb n Cyc (X, q) ⊂ Hilb β,n (X) is the following subscheme to be the open subscheme containing Cohen-Macaulay subschemes of Z.
Lemma 4.2.1. Suppose Z ⊂ X is a one dimensional Cohen-Macaulay subscheme such that: (1) Z has the decomposition Z Proof. We show the action is well defined on a flat family in Hilb n CM (X, Cyc(Z)). Let such a family be given by the diagram: The reduced curves C, Z red i and the neighbourhoods V i must all be constant on the family and we have a decomposition Hence, the action is given by Consider the constructible map where (κ * 1)(z) := e(κ −1 (z)) and the last line comes from the following lemma.
Proof. Let α ∈ (C * ) 2 and z ∈ Hilb n CM (X, q) correspond to Z ⊂ X. Also let Z = ∪Z i and V = V i be as in lemma 4.2.1 4.2.1. Then the fibre κ −1 (x) is the where the last equality is in K 0 (Var C )((p)) from lemma 4.1.2 4.1.2. Also from lemma 4.1.4 4.1.4 we have a geometrically bijective constructible morphism: We have I α·Z | X\ V = I Z | X\ V so Quot n1 X (I Z , X \ V ) ∼ = Quot n1 X (I α·Z , X \ V ). Moreover, we have isomorphisms Quot n2 X (I Z , V ) ∼ = Quot n2 X (I α·Z , V ) Taking Euler characteristic now shows that e κ −1 (x) = e κ −1 (α · x) .

4.2.3.
We will now consider a useful tool in calculating Euler characteristics of the form given in (5 5). First let z ∈ Hilb n CM (X, q) correspond to Z ⊂ X such that Z is locally monomial. Then the fibre κ −1 (x) is where the last equality is in K 0 (Var C )((p)) from lemma 4.1.2 4.1.2. To compute this fibre we employ the following method: (1) Decompose X by X = Z W where W := X \ Z (2) Let Z be set of singularities of Z red .
(3) Let i Z i = Z \ Z be a decomposition into irreducible components.
Then applying Euler characteristic to lemma 4.1.4 4.1.4 we have: 4.3. Partitions and the topological vertex. We recall the terminology of 2D partitions, 3D partitions and the topological vertex from [ORV ORV, BCY BCY]. A 2D partition λ is an infinite sequence of decreasing integers that is zero except for a finite number of terms. The size of a 2D partition |λ| is the sum of the elements in the sequence and the length l(λ) is the number of non-zero elements. We will also think of a 2D partition as a subset of (Z ≥0 ) 2 in the following way: A 3D partition is a subset η ⊂ (Z ≥0 ) 3 satisfying the following condition: (1) (i, j, k) ∈ η if and only if one of i, j or k is zero or one of (i − 1, j, k), (i, j − 1, k) or (i, j, k − 1) is also in η.
The leg of η in the ith direction is the subset {(i, j, k) ∈ η | (j, k) ∈ λ}. We analogously define the legs of η in the j and k directions. The weight of a point in η is defined to be Using this we define the renormalised volume of η by: The topological vertex is the formal Laurent series: where the sum is over all 3D partitions asymptotic to (λ, µ, ν). An explicit formula for V λµν is derived in [ORV ORV,Eq. 3.18] to be: 4.4. Partition Thickened Section, Fibre and Banana Curves. In this subsection we consider non-reduced structure for curves in our desired classes. The partition thickened structure will be the fixed points of a (C * ) 2 -action.

4.4.1.
Recall that the section ζ ∈ S is the blow-up of a point in z ∈ P 2 . Choose once and for all a formal neighbourhood Spec C[[s, t]] of z ∈ P 2 . The blow-up gives the formal neighbourhood of ζ ∈ S with 2 coordinate charts: with change of coordinates s → tv and u → v −1 . This gives the formal neighbourhood of σ ∈ X with 2 coordinate charts: with change of coordinates s i → t i v and u → v −1 . We call these coordinates the canonical formal coordinates around σ ∈ X.

4.4.5.
We now consider a canonical formal neighbourhood of the banana curve C 3 . We follow much of the reasoning from [Br Br,§5.2]. Let x ∈ S correspond to a point where π : S → P 1 is singular. Let formal neighbourhoods in the two isomorphic copies of S be given by and the map S → P 1 be given by r → s i t i . Then the formal neighbourhood of a conifold singularity in X is given by and the restriction to a fibre of the projection S × P 1 S → P 1 is Spec C[[s 1 , t 1 , s 2 , t 2 ]]/(s 1 t 1 , s 2 t 2 ). Now, blowing up along {s 1 = t 2 = 0} (which is canonically equivalent to blowing up along {s 1 − t 1 = s 2 − t 2 = 0}), we have the two coordinate charts: , and where the change of coordinates is given by t 1 → vs 2 , t 2 → vs 1 and u → v −1 . We call these coordinates the canonical formal coordinates around the banana curve C 3 .
4.4.6. With these coordinates we have: (1) Then the restriction to the fibre of pr : X → P 1 is (2) The banana curve C 3 is given by 4.4.7. Similar to 4.4.2 4.4.2 we also consider canonical relative coordinates for a C 3 banana curve. Recall 3.4.4 3.4.4 and let D is the image under ν : F ban → F ban of the proper transform under bl : Bl (0,∞),(∞,0) (P 1 ×P 1 ) → P 1 ×P 1 of a smooth divisor in |f 1 +f 2 | on |P 1 × P 1 |.
If D intersects (0, 0) then the restriction of D to the formal neighbourhood of C 3 is given by: for some a ∈ C * . In this case we define canonical formal coordinates relative to D around a C 3 banana by the following change of coordinates.

4.4.11.
It is also shown in [Br Br,§5.2] that there are the following formal coordinates on C 2 compatible with the canonical formal coordinates around C 3 : where the change on coordinates is given by s 2 → t 2 , s 1 → t 1 t 2 and v → t 2 u. We can define partition thickenings and a compatible (C * ) 2 -action in these coordinates. (1) The canonical (C * ) 2 -action on these coordinates is defined by: (s 1 , v, s 2 ) → (λ 1 s 1 , v, λ 2 s 2 ) and (t 1 , u, t 2 ) → (λ 2 t 1 , u, λ 1 t 2 ).
Remark 4.4.14. The partition thickened curves described in this section are easily shown to be the only Cohen-Macaulay subschemes supported in these neighbourhoods that are invariant under the (C * ) 2 -action. This is because the invariant Cohen-Macaulay subschemes must be generated by monomial ideals.

Relation between Quot Schemes on C 3 and the Topological Vertex.
This section is predominately a summary of required results from [ BK BK]. For 2D partitions λ, µ and ν we define the following subscheme of C 3 : where C λ,∅,∅ is defined by the idea I λ,∅,∅ := (t λ1 , . . . , t l−1 s λ l , s l ), with C ∅,µ,∅ and C ∅,∅,ν being cyclic permutations of this. Also define the ideal by I λµν = I λ∅∅ ∩ I ∅µ∅ ∩ I ∅∅ν Now we consider the Quot scheme of length n quotients that are supported at the origin and we employ the following simplifying notation: Quot n (λ, µ, ν) := Quot n C 3 (I λµν , {0}) The quotients parametrised here have kernels that are the ideal sheaf of a onedimensional scheme Z with underlying Cohen-Macaulay curve C λ,µ,ν . The embedded points of this scheme are all supported at the origin, but Z doesn't have to be locally monomial. We use the following variation of the notation for the topological vertex: Lemma 4.5.1. Let C be a partition thickened section, fibre or C 3 -banana curve thickened by λ. Then (1) If x ∈ C is a smooth point then e Quot n X (I C , {x}) = V λ∅∅ . (2) If C is a thickened nodal fibre then e Quot n X (I C , {x}) = V λλ t ∅ . Let C be a reduced curve intersecting C at y ∈ C such that I C ∩ I C is locally monomial and there are formal local coordinates C [[r, s, t]] at y such that: Proof. The proof is the same as [ BK BK] Lemma 15. (1) We have: .
(2) Let λ be a 2D partition and λC ⊂ D be either a partition thickened section, fibre or C 3 banana and let T be finite set of points on C such that C \ T is smooth. Then .
Proof. The argument is the same as that given for equation (9) in [ BK BK].
The standard (C * ) 3 -action on C 3 induces an action on the Quot schemes. The invariant ideals I ⊂ C[r, s, t] are precisely those generated by monomials. Also, since there is a bijection between locally monomial ideals and 3D partitions we see that where we are summing over 3D partitions asymptotic to (λ, µ, ν) and n(η) is the number of boxes not contained in any legs. Note that that the lowest order term in V λµν is one, which is not true about V λµν in general. In fact we have the relationship: where η min is the 3D partition associated to C λµν , and | · | is the renormalised volume defined in eqn (6 6).
Lemma 4.5.3. If λ is a 2D partition then we have the following equalities: Figure 9. Depiction of the decomposition of the Chow subscheme that parametrises the vertical fibres of pr 2 . The red dots indicate when the fibres don't intersect the section i.e. Sm ∅ 2 and N ∅ 2 . The white dots indicate when the fibres do intersect the section i.e. Sm σ 2 and N σ 2 .
Proof. Parts (1), (2) and (4) are directly from [ BK BK] lemma 17. For part 3, there are λ 1 boxes that are in the λ-leg and one of the -legs. There are λ t 1 boxes that are in the λ-leg and the other -leg. There is one box that is contained in all three so the renormalised volume is calculated to be 5. Calculating the Euler Characteristic from the Fibres of the Chow Map 5.1. Calculation for the class σ + (0, •, •). We now recall some previously introduced notation: (1) B i = {b 1 i , . . . , b 12 i } is the set of the 12 points in S i that correspond to nodes in the fibres of the projection π : are the 12 nodal fibres of π : S i → P 1 with the nodes removed and: where Sm σ i := Sm i ∩ σ and Sm ∅ i := Sm i \ σ. Now from lemma 3.5.1 3.5.1 we can further decompose Chow σ+(0,•,•) (X) as: Moreover, if q = (ax, cy, dz, lw, mb 2 , nb op ) ∈ Chow σ+(0,•,•) (X) then the fibre is given by 5.1.1. Suppose C is Cohen-Macaulay with the cycle given above. Note that C can be decomposed into a part supported on C 2 and C 3 and a part supported away from the banana configuration. This gives the following formal neighbourhoods and (C * ) 2 -actions: (1) Let U i be the formal neighbourhood of C in X. These have a canonical (C * ) 2 -action described in 4.4.8 4.4.8 and 4.4.12 4.4.12. (2) Let V i be the formal neighbourhood of pr −1 2 (y i ) in X. These have a canonical (C * ) 2 -action described in definition 4.4.4 4.4.4 and σ ∩ V i is either empty of invariant under this action.
Hence the conditions of lemma 4.2.1 4.2.1 are satisfied and there is a (C * ) 2 -action defined on Hilb n CM (X, q). Using the partition thickened notation introduced in section 4.4 4.4 we introduce the subschemes: and their ideals I α,γ,δ,λ,µ,ν in X where α, γ, δ, λ, µ and ν are tuples of partitions of a, c, d, l, m and n respectively. Then using this notation we can identify the fixed points of the action as the following discrete set: (1) Decompose X by X = W C α,γ,δ,λ,µ,ν where W := X \ C α,γ,δ,λ,µ,ν .

5.2.
Preliminaries for classes of the form •σ + (i, j, •). We recall from lemma 3.5.3 3.5.3 that there is a decomposition of Chow •σ+(i,j,•) (X) such that for any point q ∈ Chow •σ+(i,j,•) (X) the fibre is for some one dimensional subscheme C of X with where D is a one dimensional reduced subscheme of X. We see from lemma 3.5.3 3.5.3 that the intersection of D with σ has length 0, 1 or 2. We consider the following formal neighbourhoods around components of C: (1) Let U i be the formal neighbourhood of C  By construction the restrictions of D to these neighbourhoods are invariant under these actions. Hence the conditions of lemma 4.2.1 4.2.1 are satisfied and there is a (C * ) 2action defined on Hilb n CM (X, Cyc(C)). We introduce the notation for subschemes of X: and their ideals I α,µ . Then using this notation we can identify the fixed points of the action as the following discrete set: Using the result of 4.2.2 4.2.2 we have Where the holomorphic Euler characteristic χ(O Cα,µ ) is given by the following lemma.
Lemma 5.2.1. The holomorphic Euler characteristic of C α,µ is: Proof. This is immediate from the exact sequence decomposing C α,µ into irreducible components: Using the decomposition method of 4.2.3 4.2.3 we take the following steps: (1) Decompose X by X = W C α,µ where W := X \ C α,µ .
(2) Let C α,µ be set points given by the following disjoint sets: (3) Denote the components supported on smooth reduced sub-curves by:

5.2.3.
Then applying Euler characteristic to lemma 4.1.4 4.1.4 we have: We have that e(X) = 24 and e(σ) = e(C 3 ) = 2. So the Euler characteristic of W is: Hence now have from lemma 4.5.2 4.5.2 that the first two lines from above will be: The intersection of D and ασ will determine the third line. From lemma 4.5.2 4.5.2 and lemma 4.5.3 4.5.3 it will be one of: Similarly the factors of the fourth line will determined by the intersections D ∩ C (i) 3 to be (the fourth comes from 4.4.9 4.4.9): (1) p

5.2.5.
We can calculate e Hilb • Cyc (X, q) using the above results and notation from 5.2.4 5.2.4: where Φ σ and Φ i are determined by the intersections of σ and C (i) 3 respectively to be one of the following functions: 5.3. Calculation for the class •σ + (0, 0, •). From lemma 3.5.3 3.5.3 have the decomposition of Chow •σ+(0,0,•) (X) into: Recall equation (7 7) from section 5.2 5.2 and the notation: In this class we have D = ∅. Hence we have the following summary of results from 5.2.4 5.2.4 and 5.2.5 5.2.5.
Now we have: Where the last equality is from 6.3.4 6.3.4 part 2 2 and 6.3.2 6.3.2 part 1 1.

5.4.
Calculation for the class •σ + (0, 1, •). Recall the previously introduced notation: (1) B i = {b 1 i , . . . , b 12 i } is the set of the 12 points in S i that correspond to nodes in the fibres of the projection π : S i → P 1 .
are the 12 nodal fibres of π : S i → P 1 with the nodes removed and: Now from lemma 3.5.3 3.5.3 we can further decompose Chow •σ+(0,1,•) (X) into the four parts: Recall equation (7 7) from section 5.2 5.2 and the notation: Each part will be characterised by the type of D. We consider parts (1)-(4) separately to part (5).

Parts (1)-(4):
In parts (1)-(4) the curve D is a fibre of the projection pr 2 : X → S. The following table is the summary of results from 5.2.4 5.2.4 and 5.2.5 5.2.5 when applied to the particular D's arising in each strata: The union of parts (1) Which becomes: From lemmas 6.3.2 6.3.2, 6.3.4 6.3.4 and 6.3.5 6.3.5 we have: So we have:

Part (5):
We have 12 separate isomorphic strata: These parameterise when D = C 2 . The following is the summary of results from 5.2.4 5.2.4 and 5.2.5 5.2.5.
From lemmas 6.3.2 6.3.2 and 6.3.4 6.3.4 we have: ( Since the strata are isomorphic we have: Thus combining parts (1)-(5) we have that the overall formula is: Calculation for the class •σ + (1, 1, •). We have a decomposition from lemma 3.5.3 3.5.3 of Chow (1,1,•) (X) into the parts: Diag • We also recall the notation from equation (7 7) from section 5.2 5.2 and the notation: Each part will be characterised by the type of D. We will consider each case (a)-(f) separately and will use the following the previously introduced notation throughout: (1) B i = {b 1 i , . . . , b 12 i } is the set of the 12 points in S i that correspond to nodes in the fibres of the projection π : 3) N i ⊂ S i are the 12 nodal fibres of π : S i → P 1 with the nodes removed and: where Sm σ i := Sm i ∩ σ and Sm ∅ i := Sm i \ σ. We will also use the new notation: ∩ D Here we have grouped by the number and type of intersection with σ.
The contribution from grouping (4) is: Grouping (5) (5): From lemmas 6.3.2 6.3.2 and 6.3.4 6.3.4 we have: Summing the contributions from the above groupings we arrive at the overall contribution from part (a): By the symmetry of X we only need to consider part (b), with part (c) being completely analogous. For each k ∈ {1, . . . , 12} we begin by decomposing S • 1 into the following six parts: is the connected component of N σ 1 corresponding the the kth banana configuration and N σ,c 1 is its complement in N σ 1 . The same definition is true for N ∅ 1 .
We use the above size part decomposition for The following table is the summary of results from 5.2.4 5.2.4 and 5.2.5 5.2.5 for this stratification.
There are 12 singular fibres of pr. So, we have that the combined contribution from parts (c) and (d) is: (1) The following table is the summary of results from 5.2.4 5.2.4 and 5.2.5 5.2.5 for this stratification.

Part (f ):
Recall from lemma 3.5.3 3.5.3 that part (f), Diag • has the further decomposition: Where we have used the notation: (1) J 0 and J 1728 to be the subsets of points x ∈ P 1 such that π −1 (x) has j-invariant 0 or 1728 respectively and J = J 0 J 1728 .
(2) L to be the linear system |f 1 + f 2 | on P 1 × P 1 with the singular divisors removed where f 1 and f 2 are fibres of the two projection maps.

Parts (g)-(i):
The results for parts (g)-(i) will all be very similar. The key differences are: (1) The overall factor of Q 3 may be different.
From lemmas 6.3.2 6.3.2, 6.3.4 6.3.4 and 6.3.5 6.3.5 we have: The overall factors of Q n 3 are calculated in 3.3.2 3.3.2 to be: (1) n = 4 for (g) and n = 0 for (h).
So the contribution for grouping (4) is: Combining groupings (1)-(4) we have the overall contribution for part (j) is: 6. Appendix

Connected Invariants and their Partition Functions.
For the rank four sub-lattice Γ ⊂ H 2 (X, Z) generated by a section and banana curves, we can consider the connected unweighted Pandharipande-Thomas invariants. They are defined formally via the following partition function For the partition function in theorem A A we consider the first terms of the expansion in Q σ and Q 1 : So the first terms of the expansion in Q σ and Q 1 of the connected partition function are: In particular we have the connected version of Z σ+(0,•,•) as: proving corollary B B. For the partition function in theorem C C we consider the first terms of the expansion in Q 1 and Q 2 : So the first terms of the expansion in Q 1 and Q 2 of the connected partition function are: In particular we have the connected version of Z •σ+(0,0,•) as and the connected version of Z •σ+(0,1,•) (and also of Z •σ+(1,0,•) ) given by: and the connected version of Z •σ+(1,1,•) given by: Z PT,Con •σ+(1,1,•) = 12 Q 4 3 (2ψ 0 + ψ 1 ) + Q 3 3 (8ψ 0 + 6ψ 1 + ψ 2 ) + Q 2 3 (12ψ 0 + 10ψ 1 + 2ψ 2 ) + Q 3 (8ψ 0 + 6ψ 1 + ψ 2 ) + (2ψ 0 + ψ 1 ) + Q σ 12ψ 0 + 2ψ 1 + Q 3 48ψ 0 + 44ψ 1 + Q 2 3 216ψ 0 + 108ψ 1 + 24ψ 2 + Q 3 3 48ψ 0 + 44ψ 1 + Q 4 3 12ψ 0 + 2ψ 1 . Corollary D D now follows immediately. 6.2. Linear System in P 1 × P 1 . In this section we consider a stratification of the following linear system in P 1 × P 1 with strata determined by the intersections of the associated divisors with a collection of points.
Consider the fibres of the projection maps pr i : P 1 × P 1 → P 1 and a fibre from each f i . The linear system in P 1 × P 1 defined by the sum of a fibre from each map is |f 1 + f 2 | = P 3 . This is the collection of bi-homogeneous polynomials of degree (1, 1): ax 0 y 0 + bx 0 y 1 + cx 1 y 0 + dx 1 y 1 = 0 [a : b : c : d] ∈ P 3 6.2.1. There are five points in P 1 × P 1 that are of interest to us:  where we have used the standard notation 0 = [0 : 1] and ∞ = [1 : 0]. We will decompose |f 1 + f 2 | into strata based on which points the divisor intersects. Consider a divisor D ∈ |f 1 + f 2 |. Then D passes through: 6.2.2. Define the following convenient notation for y, x ∈ P: (1) Sing ⊂ |f 1 + f 2 | is the subset of singular divisors.
(2) L ∅ ⊂ |f 1 + f 2 | \ Sing is the subset of smooth curves not passing through any points of P . (3) L x ⊂ |f 1 + f 2 | \ Sing is the subset of smooth curve passing through x but no other points of P . (4) L x,y ⊂ |f 1 + f 2 | \ Sing is the subset of smooth curve passing through x and y but no other points of P. (5) Also let L σ ∅ , L σ x and L σ x,y be subsets of L ∅ , L x and L x,y respectively with the further condition that the curves pass through σ. (6) Let L ∅ ∅ , L ∅ x and L ∅ x,y be the complements of L σ ∅ , L σ x and L σ x,y in L ∅ , L x and L x,y respectively. With this notation we have the following decomposition of |F 1 + F 2 |: ax 0 y 0 + bx 0 y 1 + cx 1 y 0 + dx 1 y 1 = (αx 0 + βx 1 )(γy 0 + δy 1 ) = 0 where [α : β], [γ : δ] ∈ P 1 . Hence Sing ∼ = P 1 ×P 1 and the Euler characteristic is e(ban) = e(|f 1 + f 2 |) = 4. L x,y : We consider for x = (0, 0) and y = (∞, ∞) with the case (0, ∞) and (∞, 0) being completely analogous. The points [a : b : d : c] ∈ |f 1 + f 2 | correspond to a curve passing through x and y if and only if a = d = 0. Moreover, this is singular when either b = 0 or c = 0. Hence L x,y ∼ = P 1 \ {0, ∞} and e(L x,y ) = 0.
The set L σ x,y is when b+c = 0, which is a point in P 1 . So we have e(L σ x,y ) = 1 and e(L ∅ x,y ) = 1. L x : We consider the case x = (0, 0) with the other cases being completely analogous. So the subspace of all divisors passing through x is [a : b : d : c] ∈ |f 1 + f 2 | where d = 0. This is a P 2 ⊂ P 3 . The subspace where the curve doesn't pass through one of the other points is where a, b, c = 0 which is given by C * × C * ∼ = P 2 \ {a = 0} ∪ {b = 0} ∪ {c = 0} . None of the equations for these curves factorise since such a factorisation would require either b = 0 or c = 0. Hence, L x ∼ = C * × C * and e(L x ) = 0.
The subset L σ x is defined by the further condition a + b + c = 0 which gives L σ x = [a : b : c] ∈ P 2 a, b, c = 0 and a + b = 1 ∼ = C * \ pt Hence we have the Euler characteristics e(L σ x ) = −1 and e(L ∅ x ) = 1. L ∅ : The set of curves not passing through any points of P is given by The singular curves are given by the factorisation condition: x 0 y 0 + bx 0 y 1 + cx 1 y 0 + dx 1 y 1 = (x 0 + βx 1 )(y 0 + δy 1 ) which is the condition that d = bc. So the subspace of curves which are singular is (C * ) 2 ⊂ (C * ) 3 . Hence L ∅ ∼ = {(b, c, d) ∈ (C * ) 3 |b = dc} and e(L ∅ ) = 0.