A Remark on Gromov-Witten-Welschinger 3 3 Invariants of Cp #Cp

A REMARK ON GROMOV-WITTEN-WELSCHINGER 3 3 INVARIANTS OF CP #CP

YANQIAO DING

Abstract. We generalize the formula of Gromov-Witten-Welschinger 3 invariants of CP established by E. Brugall´eand P. Georgieva in [BG16b] 3 3 to CP #CP . Using pencils of quadrics, some real and complex enu- 3 3 merative invariants of CP #CP can be written as the combination of 2 the enumerative invariants of the blow up of CP at two real points.

1. Introduction It is a long history for mathematicians to find appropriate way to count curves. Almost 24 years ago, inspired by string theory [Wit91], Y. Ruan and G. Tian introduced the definition of Gromov-Witten invariants on semi- positive symplectic manifolds [RT95]. Then the definition of Gromov-Witten invariants was generalized to symplectic manifolds [Rua99, FO99, LT98b] and algebraic varieties [LT98a, Beh97]. With the help of Gromov-Witten invariants, complex enumerative geometry experiences a rapid development. The emergence of real enumerative invariants, such as Welschinger invariants [Wel05a, Wel05b], real Gromov-Witten invariants [GZ18], and refined tropical enumerative invariants [BG16a], also prompts the development of real enumerative geometry significantly. Welschinger invariant is a sign counting of real curves in real rational surfaces or real convex algebraic threefolds. There are many methods, for example, tropical geometry and degeneration formula, to investigate the Welschinger invariants of real rational surfaces [IKS09, IKS13, IKS15, BP15, Bru18, BM07, Bru15, Bru18, DH18, Din17]. However, the methods for com- putation of Welschinger invariants of threefolds are limited. For instance, some particular cases of Welschinger invariants of threefolds were calculated arXiv:1902.07486v1 [math.AG] 20 Feb 2019 in [BM07, GZ17, Wel07, PSW08]. Kollárproposed pencils of quadrics of 3 3 CP can be used to compute the enumerative invariants of CP [Kol15]. E. Brugalléand P. Georgieva completely computed the Welschinger invariants 3 of CP [BG16b]. They estiblished a relation between the GWW invariants 3 1 1 of CP and the GWW invariants of CP × CP . In this note, we generalize 3 3 this relation to CP #CP .

2010 Mathematics Subject Classiﬁcation. Primary 14P05, 14N10; Secondary 14N35, 14P25. Key words and phrases. Real enumerative geometry, Welschinger invariants, Gromov- Witten invariants. 1 2 YANQIAO DING

3 1 Equip CP and CP with the standard real structure which is the complex 3 3 3 conjugation. Denote by Y = CP #CP the blow-up of CP at a real point 3 x0, by l the strict transform of a line in CP which does not pass through 1 1 x0, by E the exceptional divisor, and by Q = CP × CP a quadric surface 3 in CP passing through x0. According to the classification results of real surfaces [Kol97, DK00, DK02], the blow-up at a real point x0 of the quadric 1 1 surface Qe = (CP ×CP )1,0 is real deformation equivalent to the blow up of 2 2 CP at two real points CP2,0. Let l1, l2 be the strict transforms of two lines, 1 1 which are CP × {p0} and {p0} × CP up to interchanging with each other, 1 1 1 1 in CP × CP not passing x0. The homology group H2((CP × CP )1,0; Z) is generated by the classes [l ], [l ] and [E ], where E is the exceptional 1 2 Qe Qe 1 1 divisor of (CP × CP )1,0. We use (a, b, c) as an abbreviation of the class a[l ] + b[l ] − c[E ] in H (( P 1 × P 1) ; ). 1 2 Qe 2 C C 1,0 Z Let d and k be two positive integers with d ≥ k. Given a generic real 3 3 configuration x of 2d−k points in CP #CP which contains s pairs of complex conjugated points. Denote by W 3 (d[l] − k[e], s) the Welschinger CP 3#CP invariant “counting” the real rational curves passing through x and repre- 3 3 senting the class d[l]−k[e] in CP #CP , where e is a line in the exceptional divisor E. Denote by W ((a, b, k), s) the Welschinger invariant “counting” Qe the real rational curves passing through y, which is a generic real configuration of 2(a + b) − k − 1 points in Qe with s pairs of complex conjugated points, and representing the class (a, b, c) in Qe.

Theorem 1.1. For any d ∈ Z>0 − 2Z, k ∈ 2Z≥0 with d ≥ k, and s ∈ k {0, . . . , d − 2 − 1}, we have

X a+ k (1) W 3 (d[l] − k[e], s) = (−1) 2 (d − 2a)W ((a, b, k), s). CP 3#CP Qe a+b=d 0≤a

By Remark4 .4, W 3 (d[l]−k[e], s) = 0, where d, k are positive even CP 3#CP k integers and s ≤ d − 2 − 1. Since Welschinger invariants of any blow-up of 3 the real projective plane and the real quadric Q in CP are computed by Horev and Solomon in [HS12], one can use equation (1) to determine all the Welschinger invariants W 3 (d[l] − k[e], s) with k is even. CP 3#CP A similar result of Gromov-Witten invariants can also be derived. Denote 3 3 CP #CP by h[pt],..., [pt]i0,d[l]−k[e] the Gromov-Witten invariant counting the rational 3 3 curves in CP #CP passing through 2d − k generic points and representing Qe the class d[l] − k[e]. Denote by h[pt],..., [pt]i0,(a,b,k) the Gromov-Witten invariant counting the rational curves in Qe passing through 2(a + b) − k − 1 generic points and representing the class (a, b, c). REMARK ON GW-W INVARIANTS 3

Theorem 1.2. Let d ∈ Z>0, k ∈ Z≥0 and d ≥ k. Then

3 3 CP #CP X 2 Qe (2) h[pt],..., [pt]i0,d[l]−k[e] = (d − 2a) h[pt],..., [pt]i0,(a,b,k). a+b=d 0≤a

Acknowledgments: The author would like to thank Jianxun Hu for his continuous support and encouragement as well as enlightening discussions.

2. Welschinger invariants 2 2.1. Welschinger invariants of CP2,0. Given a, b, c ∈ Z≥0 with 2(a+b)− 1 1 c − 1 > 0. Let Qe = (CP × CP )1,0 which is real deformation equivalent to 2 CP2,0 and x ⊂ Qe be a real configuration of 2(a+b)−c−1 points with s pairs of complex conjugated points. Denote by C(x) the set of real rational curves representing the class (a, b, c) and passing through x in Qe. If x is generic enough, the set C(x) is finite, and every curve C ∈ C(x) is an immersion with only nodal points. Denote by m (C) the number of real isolated nodes Qe (nodes with two complex conjugated branches) of C. The integer X m (C) W ((a, b, c), s) = (−1) Qe Qe C∈C(x) is independant of the choice of the real configuration x, and it only depends on a, b, c, and the number s of pairs of complex conjugated points in x (see [Wel05a]).

3 3 2.2. Welschinger invariants of CP #CP . Given d, k ∈ Z≥0 with d ≥ k, and let x be a generic real conﬁguration, which contains s pairs of complex conjugated points and at least one real point, of 2d−k points in Y . Denote by C0(x) the set of irreducible real rational curves in Y of class d[l]−k[e] passing 0 1 0 through x. The set C (x) is ﬁnite, and every element f : CP → Y of C (x) is ∗ 1 a balanced curve, i.e. f is an immersion and the quotient Nf = f T Y/T CP is isomorphic to the holomorphic bundle OCP 1 (2d−k−1)⊕OCP 1 (2d−k−1). 3 3 Fix a spin structure s on RP and a spin structure ¯s on RP . They can 3 3 induce a spin structure s#¯s on RY = RP #RP . Then the integer

X sY (C) WY (d[l] − k[e], s) = (−1) C∈C0(x) is independant of the choice of x, where sY (C) is the spinor state of C. It depends only on the class d[l] − k[e], the number s of complex conjugated pairs in x and the spin structure s#¯s (see [Wel05b]). Since we only deal with the case when k is even in this note, we always assume k is even. The spinor state sY (C) is deﬁned as follows. 1 0 Let (f : CP → Y ) ∈ C (x) be a balanced real algebraic immersion. 3 3 Fix an orientation on RY = RP #RP and equip RY with a Riemannian 4 YANQIAO DING

1 metric g. Choose an orientation on f(RP ) which is induced by the orientation on P 1. The tangent bundle T P 1 is a subbundle of f ∗ T Y . R R |RP 1 R The real vector bundle RNf , which is isomorphic to the orthogonal subbundle of T P 1 in f ∗ T Y , has an induced orientation such that a R |RP 1 R 1 positive orthonormal frame of TxRP followed by a positive orthonormal ∗ frame of RNf |x provides a positive orthonormal frame of fx T RY for ev- 1 ery x ∈ RP . Denote by H the class of the real part of a section of ∼ 1 1 P (Nf ) = CP × CP having vanishing self-intersection, by F the class of a real fiber of P (RNf ). The orientations on H, and F are induced by the 1 orientations on RP and RNf respectively. Choose a real holomorphic line subbundle Lf ⊂ Nf such that P (Lf ) ⊂ P (Nf ) is a section of bidegree (1, 1) whose real part is homologous to ±(H + F ) ∈ H1(RNf ; Z). The bundle 1 T RP ⊕ RLf is equipped with a Riemannian metric induced by the one of T RY . Choose a loop of positive orthonormal frames (σ1(x), σ2(x))x∈RP 1 1 of T RP ⊕ RLf such that (σ1(x))x∈RP 1 is a loop of positive orthonormal 1 ∗ frames of T P . Let (σ3(x)) 1 be the unique section of f 1 T Y such R x∈RP |RP R that (σ1(x), σ2(x), σ3(x))x∈RP 1 is a loop of positive orthonormal frames of ∗ f 1 T Y . Define SY (f) = 0 if this loop of the SO3( )-principal bundle RY |RP R R of orthonormal frames of T RY lifts to a loop of the associated Spin-principal bundle PY , and SY (f) = 1 otherwise. Note the spinor state depends only on the spin structure s#¯s and the isotopy class of RLf . A more geometric way to explain the construction of the holomorphic line bundle Lf is as follows. The choice of the class ±(H + F ) is equivalent to 1 choice a direction to rotate the ribbon of f(RP ) provided by the real holomorphic section H ∈ P (Nf ) such that the ribbon after rotated is orientable. The isotopy class of RLf is the isotopy class of the real part of the degree 2d−k−2 line subbundle of Nf whose real fibers rotate positively with respect to the positive basis of RNf in local holomorphic coordinates on Nf defining a real holomorphic splitting Nf = OCP 1 (2d − k − 1) ⊕ OCP 1 (2d − k − 1) (see [BG16b, Section 2.2] and [Wel05b, Section 2.2] for more details).

3 3. Rational curves on a blow-up of smooth quadric of CP 3.1. Elliptic curves, pencils of quadrics and blow-up. Let C be a complex elliptic curve. The choice of a point p in C will induce an isomorphism Φ between the elliptic curve C and the group Pic0(C). This isomorphism gives a group structure on C. If C is considered as the quotient of C by a full rank lattice Λ for which p is identiﬁed with 0, the group structure induced by Φ is the same with the group structure induced from (C, +) by the quotient map. Φ also induces a series of isomorphisms Φd between Picd(C) and C (see [BG16b, Section 3.1] for more details). In this note, Picd(C) is always regarded as C under the isomorphism Φd. If C is also real, with RC 6= ∅, and p ∈ RC, there are two cases as follows (see [BG16b, Section 3.1]): REMARK ON GW-W INVARIANTS 5

• RC has two connected components: if d is even, both components contain d torsion points of order d on C; if d is odd, the connected component of RC containing p contains d torsion points of order d; • RC is connected, and RC contains d torsion points of order d; 3 Let C4 be a non-degenerate elliptic curve of degree 4 in CP , and Q be 3 the pencil of quadrics in CP determined by C4. Let h ∈ C4 ' Pic4(C4) be the hyperplane section class. There is a ramiﬁed covering map πQ : C4 ' Pic2(C4) → Q of degree two. From analysing the ramiﬁcation value of π, one can get Q contains 4 singular quadrics. If C4 is real and RC4 6= ∅, there are three cases as follows (see [BG16b, Section 3.2]):

• RC4 is not connected, and h, p are not on the same component: there is no real singular quadric in Q; • RC4 is not connected, and h, p are on the same component: there are 4 real singular quadrics in Q; • RC4 is connected: there are 2 real singular quadrics in Q; 3 3 3 3 Let π : Y = CP #CP → CP be the blow-up of CP at x0 ∈ C4. Denote by Ce4 ⊂ Y and Qe ⊂ Y the strict transforms of C4 and Q respectively. 3 Denote by H the strict transform of a hyperplane in CP which does not pass x0, and by h˜ = H| ∈ Pic4(C4) the hyperplane section class. The Ce4 e map π : C ' Pic (C ) → Q induces a ramiﬁed covering map π : C ' Q 4 2 4 Qe e4 Pic (C ) → Q of degree two, where π (x), x ∈ C , is the strict transform of 2 e4 e Qe e4 πQ ◦ π (x). If C4 is real, with C4 6= ∅, and p, x0 ∈ C4, there are three |Ce4 R R cases as follows: ˜ • RCe4 is not connected, and h, p are not on the same component: there is no blow-up of real singular quadric in Qe; ˜ • RCe4 is not connected, and h, p are on the same component: there are 4 blow-up of real singular quadrics in Qe; • RCe4 is connected: there are 2 blow-up of real singular quadrics in Qe;

3.2. Rational curves on blow-up of smooth quadrics. Let x ⊂ C4 \x0 be a conﬁguration of 2d − k distinct points, and e = E | . Denote by Qe Qe Ce4 0 C (x) the set of connected algebraic curves of arithmetic genus 0 representing d[l] − k[e] in Y and passing through x.

Lemma 3.1. Suppose the points in x are in general position in C4 \ x0. 0 Then every curve C in C (x) is irreducible and is contained in a blow-up of quadric of Qe. Furthermore, the blow-up of quadrics of Qe which contain a 0 curve in C (x) correspond to the solutions L2 ∈ Pic2(Ce4) of the equation (3) (d − 2a)L = (d − a)h˜ − ke − x, 2 Qe d with 0 ≤ a < 2 . 6 YANQIAO DING

0 A curve C in C (x) is linearly equivalent in such a blow-up of quadric Qe to al +(d−a)l −kE , where l (resp. l ) is a line in Q whose intersection with 1 2 Qe 1 2 e ˜ Ce4 is L2 (resp. h − L2). Any irreducible rational curve C in Qe representing 0 al + (d − a)l − kE and passing through 2d − k − 1 points of x is in C (x). 1 2 Qe Proof. The proof is similar to the proof of [Kol15, Proposition 3]. For the 0 completeness, we give it in the following. Suppose the curve C ∈ C (x) is irreducible. Since any point of C \ x is contained in some Qe, C and Qe intersect at ≥ 2d − k + 1 points which implies that C ⊂ Qe. Hence C is linearly equivalent to al + (d − a)l − kE for some 0 ≤ a ≤ d . So 1 2 Qe 2 ˜ x = (Ce4 · C) ∼ aL2 + (d − a)(h − L2) − ke (4) Qe = (d − a)h˜ − (d − 2a)L − ke . 2 Qe For the points in x are in general position, the case a = d/2 is excluded and Qe is the blow-up of a smooth quadric. 0 In the case C ∈ C (x) is reducible, let ΣCi be a connected curve passing through x representing d[l]−k[e]. If there is a component Ci passes through i more than 1/2c1(Y ) · [C ] points of x, it intersects with every Qe of Qe at i i > 1/2c1(Y ) · [C ] points. Therefore, C = Ce4 which is a contradiction. So i i i every component C passes through exactly 1/2c1(Y ) · [C ] points x ⊂ x. Since Σ[Ci] = d[l] − k[e], x is the union of the disjoint sets xi. If there are two curves Ci, Cj contained in different blow-up of quadrics i j i j i Qe 6= Qe , we have C ∩ C ⊂ Ce4. However, we know different curves C pass through different points xi. The curve ΣCi has to be disconnected which is a contradiction. Therefore, the curve ΣCi is contained in one blow-up of quadric Qe. Suppose Ci is linearly equivalent to ail + (di − ai)l − kie , we have 1 2 Qe xi ∼ (di − ai)h˜ − (di − 2ai)L − kie . 2 Qe Combined with equation (4), we obtain (d − 2a)xi ∼(di − 2ai)x + [(d − 2a)(di − ai) − (di − 2ai)(d − a)]h˜ (5) + [(di − 2ai)k − (d − 2a)ki]e . Qe We can choose the points of x such that x, h˜ and e generate a free subgroup Qe i of rank 2d − k + 2 in Pic(Ce4). So equation (5) is impossible unless x = x. Therefore, there is no reducible curve C for a general choice of the points x. 3 1 3 Given a quadric Q of CP , and let f : CP → CP be an algebraic immersion whose image is contained in Q passing through x0. Suppose z = −1 1 0 ∗ 1 ∗ 3 1 f (x0) ⊂ CP . Let Nf = f T Q/T CP , Nf = f T CP /T CP , and NQ = 3 3 ˜ 1 T CP |Q/T Q be the normal bundle of Q in CP . Let f : CP → Y be the REMARK ON GW-W INVARIANTS 7 strict transform of f. Denote by N 0 = f˜∗T Q/T P 1, N = f˜∗T Y/T P 1, f˜ e C f˜ C and N = TY | /T Q the normal bundle of Q in Y . We can get an exact Qe Qe e e 1 sequence of holomorphic vector bundles over CP : (6) 0 → N 0 → N → f˜∗N → 0. f˜ f˜ Qe If Q and f are both real, we can also obtain an exact sequence fitted by the real bundles N 0 , N and N . R f˜ R f˜ R Qe 3 The morphism π : Y → CP induces morphisms between the bundles associated to f˜ and f which are isomorphisms everywhere except z and vanish at z. Since f is an immersion, one can see the vanishing order is 1 in 0 0 local coordinates. Therefore, N = N ⊗ O 1 (−z), N = N ⊗ O 1 (−z), f˜ f CP f˜ f CP ∗ ∗ and f˜ N = f N ⊗ O 1 (−z). It follows that if f is balanced, so is f˜. Qe Q CP From [BG16b, Proposition 4.1], we know that if f has bidegree (a, b) with a 6= b, then f is balanced. So the strict transform of the algebraic immersion 1 3 f : CP → CP which is contained in Q with bidegree (a, b), a 6= b, passing through x0 is also balanced. 0 Let RNi be the real part of the normal bundle of li in Q which is a 3 subbundle of the real part of its normal bundle in CP . From [BG16b, 0 0 Proposition 4.2], one has sCP 3 (RN1) 6= sCP 3 (RN2). Following [BG16b], 0 (l1, l2) is called a positive basis of H2(Q; Z) if sCP 3 (RN1) = 0. 1 3 Proposition 3.2. Let f : CP → CP be a real algebraic immersion, whose image is contained in Q, passing through x0 such that it has bidegree (a, b) −1 1 in the positive basis with a 6= b. Suppose z = f (x0) ⊂ CP contains k points, and k is even. Let f˜ be the strict transform of f in Y . Then the holomorphic real line subbundle N 0 of N realises the real isotopy class R f˜ R f˜ RLf˜ if and only if a > b.

Proof. Recall that the isotopy class RLf˜ is constructed as follows: in the local holomorphic coordinates on Nf˜ deﬁning a real holomorphic splitting Nf˜ = OCP 1 (2d − k − 1) ⊕ OCP 1 (2d − k − 1), the isotopy class of RLf˜ is the isotopy class of the real part of the degree 2d − k − 2 line subbundle of

Nf˜ whose real fibers rotate positively with respect to the positive basis of RNf˜. Note that there are two holomorphic real line subbundles of Nf˜ of degree 2d − k − 2 depending on whether the real part of a fiber rotates in 0 0 N positively. Since N = N ⊗ O 1 (−z) and N = N ⊗ O 1 (−z), we R f˜ f˜ f CP f˜ f CP 0 can get that the fibers of RNf rotate positively with respect to the positive basis of N if and only if the fibers of N 0 rotate positively with respect R f R f˜ to the positive basis of RNf˜. From [BG16b, Proposition 4.4], we know the 0 holomorphic real line subbundle RNf of RNf realises the real isotopy class RLf if and only if a > b. 1 3 Lemma 3.3. Let f : CP → CP be a real algebraic immersion, whose image is contained in Q, passing through x0 such that it has bidegree (a, b) 8 YANQIAO DING

−1 1 in the positive basis with a 6= b. Suppose z = f (x0) ⊂ CP contains k points, and k is even. Let f˜ be the strict transform of f in Y . Then k s ( N 0 ) = m (f˜( P 1)) + b + . Y R f˜ Qe C 2 Proof. The idea of the proof is similar to the proofs of [BG16b, Lemma 2.1, Corollary 4.3] and [Wel05b, Theorem 4.4]. Let (σ (x), σ (x), σ (x)) 1 2 3 x∈f˜(RP 1) be a loop in the SO3(R)-principal bundle RY of orthonormal frames of 1 T RY constructed in Section2 .2. Fix an orientation of RP which in- ˜ 1 duces an orientation of the normal bundle RNf˜. Assume f(RP ) ∩ RE = {p1, p2, . . . , pr}. Since k is even, r must be even too. Construct a loop (˜σ (x), σ˜ (x), σ˜ (x)) of R as the concatenation of the 2r following 1 2 3 x∈f˜(RP 1) Y paths. For the convenience, let p0 = pr be the same point. When i = 0, ﬁrst, choose (σ (x), σ (x), σ (x)) . Then construct a path com- 1 2 3 x∈[p2i,p2i+1)] pletely included in the ﬁber of RY over p2i+1. This path is obtained from (σ1(p2i+1), σ2(p2i+1), σ3(p2i+1)) by having this frame turning of half a twist in the positive direction around the axis generated by σ1(p2i+1). Note that the end point of this path is the frame (σ1(p2i+1), −σ2(p2i+1), −σ3(p2i+1)).

Next, the piece is chosen to be the path (σ1(x), −σ2(x), −σ3(x))x∈[p2i+1,p2i+2]. Last, we construct a path completely included in the fiber of RY over p2i+2. This piece is obtained from (σ1(p2i+2), −σ2(p2i+2), −σ3(p2i+2)) by having this frame turning of half a twist in the negative direction around the axis generated by σ1(p2i+2). Note that the end point of this frame r is (σ1(p2i+2), σ2(p2i+2), σ3(p2i+2)). For i = 1,..., 2 − 1, we just repeat the above construction. Finally, we constructed a loop in RY which is homotopic 3 to (σ (x), σ (x), σ (x)) . Y = P 3# P is the connected sum of 1 2 3 x∈f˜(RP 1) R R R 3 3 RP and RP in a neighborhood of x0. Let the radius of the ball used to 1 construct the connected sum converge to zero, then the curve f˜(RP ) will 1 3 degenerate to the union of the curve f(RP ) and r lines RD1,...,RDr of RP passing through x . The loop (˜σ (x), σ˜ (x), σ˜ (x)) degenerates to 0 1 2 3 x∈f˜(RP 1) the union of a loop (˜σ1, σ˜2, σ˜3)f(RP 1) of RCP 3 and r loops (˜σ1, σ˜2, σ˜3)RD1 , ..., (˜σ1, σ˜2, σ˜3)RDr of RCP 3 . From the above construction, we know the loops (˜σ1, σ˜2, σ˜3)RD2i differ from (˜σ1, σ˜2, σ˜3)RD2i+1 by a non-trivial loop in a fiber of P P 3 . So s 3 ( L D ) + s 3 ( L D ) is odd. Therefore, C CP R R 2i CP R R 2i+1 r r Σi=1s 3 (RL Di ) = mod 2. CP R 2 1 For the loop (˜σ1, σ˜2, σ˜3)f(RP 1), we first smooth each node of f(RP ) as 1 Figure1, then smooth the order r singular point of f(RP ) at x0 as Figure 1 1 2. We get a collection γ of n disjoint oriented circles embedded in RP ×RP . And r n = 1 + t + mod 2, 2 1 where t is the number of hyperbolic nodes of f˜(CP ). The loops of γ do not intersect with each other, so the non-trivial loops can only realize the REMARK ON GW-W INVARIANTS 9

99K

Figure 1. Smoothing the node

99K 99K 99K

(a). order r (b). order r − 2 (c). order r − 4 (d). smoothing

Figure 2. Smoothing the order r singular point same class p[Rl1] + q[Rl2] with p and q relatively prime. The loop realising the trivial class deﬁnes a non-trivial loop in π1(SO3(R)), and the non-trivial loop deﬁnes q times the non-trivial loop in π1(SO3(R)), because (l1, l2) is 1 0 a positive basis. Suppose f(RP ) = mp[Rl1] + mq[Rl2]. So sCP 3 (RNf ) = n − m + qm mod 2. It means

0 r s 3 ( N ) = t + 1 + − m + qm mod 2. CP R f 2 ˜ 1 k2 k From the adjunction formula, f(CP ) has exactly (a − 1)(b − 1) − 2 + 2 nodes which contains t hyperbolic nodes and m (f˜( P 1)) elliptic nodes. Qe C Thus, 2 0 k k s 3 ( N ) =(a − 1)(b − 1) − + CP R f 2 2 r + m (f˜( P 1)) + 1 + − m + qm mod 2. Qe C 2 Note that k is even and (mp, mq) = (a, b) mod 2. So m is even if and only if both a and b are even. Hence

0 k 1 r s 3 ( N ) = + m (f˜( P )) + + b mod 2. CP R f 2 Qe C 2 0 0 r Since sY ( N ) = s P 3 ( N ) + Σ s 3 ( L D ) and r is even, we get R f˜ C R f i=1 CP R R i k s ( N 0 ) = m (f˜( P 1)) + b + . Y R f˜ Qe C 2 10 YANQIAO DING

4. Proof of the main results ∗ Denote by M0,n(X, d) the space of genus 0 stable maps 1 f :(CP , x1, . . . , xn) → X 1 from CP with n marked points whose images represent the class d in X. Note that the stable maps are considered up to reparametrization. There is a natural map: ∗ n ev : M0,n(X, d) → X f 7→ (f(x1), . . . , f(xn)),

n 3 3 n which is called the evaluation map. Let Ωn ⊂ Ce4 ⊂ (CP #CP \ E) be the set of conﬁgurations of n distinct points on Ce4. Let Qe ⊂ Qe be the blow-up of a quadric, a ∈ {0, . . . , d} and y ∈ Ω2d−k−1. Denote by C (y, a, b, k) the set of stable maps f :(C, x , . . . , x ) → Q, Qe 1 2d−k−1 e where C is a connected nodal curve of arithmetic genus 0, such that f(C) represents the class (a, b, k) and f({x1, . . . , x2d−k−1}) = y. 0 Proposition 4.1. There exists a dense open subset Ω2d−k−1 ⊂ Ω2d−k−1 0 such that for any a ∈ {0, . . . , d}, y ∈ Ω2d−k−1, and for every stable map (f :(C, x , . . . , x ) → Q) ∈ C (y, a, b, k), we have: 1 2d−k−1 e Qe • C is non-singular, • f is an immersion. 0 Qe For any y ∈ Ω2d−k−1, Qe contains exactly h[pt],..., [pt]i0,(a,b,k) rational curves of class (a, d − a, k) and passing through y.

Proof. Let H be the set of irreducible nodal rational curves in Qe of class 1 1 ∗ (a, d − a, k). By the proof of Lemma4 .2, we know H (CP , f T Qe) = 0. Therefore, H is a quasi-projective subvariety of dimension 2d − k − 1 of the linear system |al + (d − a)l − kE |. Let H be the Zariski closure of H and 1 2 Qe F = |(al1 +(d−a)l2 −kE )| |. In Q, C4 represents class 2[l1]+2[l2]−[E ], Qe Ce4 e e Qe so the degree of linear system F is 2d − k. As the genus of Ce4 is 1, every element of F is determined by its 2d − k − 1 points. Therefore, an element y of Ω2d−k−1 induces an element [y] of F. Ce4 is not a component of any curve in H, we can deﬁne a map φ : H → F

C 7→ C ∩ Ce4. Note that dim φ(H\H) ≤ 2d − k − 2. The image of every element f ∈ C (y, a, b, k), with y ∈ Ω such that [y] ∈/ φ(H\H), is a nodal irre- Qe 2d−k−1 1 ducible rational curve, i.e. C = CP and f is an immersion. The rest of the proof follows from Lemma4 .2. REMARK ON GW-W INVARIANTS 11

∗ Lemma 4.2. A map f ∈ M0,2d−k−1(Q,e (a, d − a, k)) is regular for the corresponding evaluation map ev if and only if it is an immersion. Proof. From the proof of [Wel05b, Lemma 1.3], we know the cokernel of d(f,y)ev is identiﬁed with the cokernel of the following composition:

0 1 ∗ 0 1 0 1 H (CP ; f T Qe) → H (CP ; Nf ) → H (CP ; Nf |y) (7) v → [v] → [v(y)], where Nf is the quotient sheaf, and the ﬁrst morphism of equation (7) is surjective. Let Nf,−y = Nf ⊗OCP 1 (−y). The short exact sequence of sheaves 0 → Nf,−y → Nf → Nf |y → 0 implies the long exact sequence 0 1 0 1 1 1 · · · → H (CP ; Nf ) → H (CP ; Nf |y) → H (CP ; Nf,−y) (8) 1 1 → H (CP ; Nf ) → 0 → · · · . 1 1 If f is an immersion, we know Nf 'OCP 1 (2d−k−2) and H (CP ; Nf ) = 0. 1 ∗ Actually, from the exact sequence 0 → T CP → f T Qe → Nf → 0, we also 1 1 ∗ can get H (CP ; f T Qe) = 0. So the cokernel of 1 1 1 1 d(f,y)ev = H (CP ; Nf,−y) = H (CP ; OCP 1 (−1)) = 0.

Now suppose f is a regular point of d(f,y)ev, then 1 1 1 1 H (CP ; Nf,−y) = H (CP ; Nf ). 1 1 1 1 1 From the cohomology of CP , we obtain H (CP ; Nf,−y) = H (CP ; Nf ) = 0. Therefore, Nf 'OCP 1 (a) with a ≥ 2d−k−2. By the adjunction formula, a ≤ 2d − k − 2. So Nf 'OCP 1 (2d − k − 2), and f is an immersion. Remark 4.3. By using [Wel05b, Lemma 4.3], one can prove, similar to the proof of [BG16b, Proposition 5.3], that the regular values of

∗ 3 3 3 3 2d−k ev : M0,2d−k(CP #CP , d[l] − k[e]) → (CP #CP ) 0 contained in Ω2d−k form a dense open subset Ω2d−k ⊂ Ω2d−k and the set 3 3 0 0 CP #CP C (x), x ∈ Ω2d−k, contains exactly h[pt],..., [pt]i0,d[l]−k[e] elements. 0 0 Proof of Theorem 1.2. Choose x ∈ Ω2d−k, where Ω2d−k is the set deﬁned in Remark4 .3, and y ⊂ x a set consisting of 2d − k − 1 points. Denote by Ma the set of blow-up of quadrics in Qe corresponding to a solution of equation (3) which consists of (d − 2a)2 elements. From Lemma3 .1 , we have 0 X X |C (x)| = |C (y, a, b, k)|. Qe 0≤a

3 0 CP #CP 3 According to Remark4 .3, |C (x)| = h[pt],..., [pt]i0,d[l]−k[e] . We can get ev- 1 3 3 0 ery map (f : CP → CP #CP ) ∈ C (x) is an immersion with the help 12 YANQIAO DING of [Wel05a, Lemma 1.3, Lemma 4.3], for x is a regular value of the corresponding evaluation map ev. By Proposition4 .1, we have |C (y, a, b, k)| = Qe Qe h[pt],..., [pt]i0,(a,b,k).

3 Proof of Theorem 1.1. Let C4 be a real elliptic curve of degree 4 in CP 3 and x0 ∈ RC4. Denote by Y the blow-up of CP at x0, and by Ce4 the strict transform of C4. Let d ∈ Z>0 − 2Z, k ∈ 2Z≥0 with d ≥ k. Choose 0 a real conﬁguration x ∈ Ω2d−k of 2d − k points which contains at least one real point. From Lemma3 .1 , we know the image of every real map 1 3 3 0 (f : CP → CP #CP ) ∈ C (x) represents class (a, d − a, k) or (d − a, a, k), d−1 where a ∈ {0, ..., 2 }, in the positive basis of the blow-up of quadric Qe 1 containing f(CP ). From Proposition3 .2 and Lemma3 .3, we can obtain ( 1 k m (f(CP )) + , if a is even, s (f( P 1)) = Qe 2 Y C m (f( P 1)) + k + 1, if a is odd. Qe C 2 0 The sum of spinor states of real elements of C (x) whose images are contained k 1+ k in Q is (−1) 2 W ((a, b, k), s) if a is even and (−1) 2 W ((a, b, k), s) if a is e Qe Qe odd. Since equation (3) always has a real solution and two real solutions diﬀer by a real torsion element of order d − 2a, we know equation (3) has d−1 d − 2a real solutions for every a ∈ {0,..., 2 }. Then one can complete the rest part of the proof with the same argument of the proof of Theorem 1.2.

Remark 4.4. The vanishing of W 3 (d[l] − k[e], s) for positive even CP 3#CP k ˜ d, k with d ≥ k and s ≤ d − 2 − 1 can also be proved in this way. If h and e are on the same component of C with p and x is on the component Qe R e4 which does not contain p, then equation (3) has no real solution.

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