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Gromov-Witten Theory of P^1 X P^1 X P^1 Dagan Karp Harvey Mudd College

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Gromov-Witten Theory of P^1 X P^1 X P^1 Dagan Karp Harvey Mudd College Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 6-18-2012 Gromov-Witten Theory of P^1 X P^1 X P^1 Dagan Karp Harvey Mudd College Dhruv Ranganathan Yale University Recommended Citation D. Karp, D. Ranganthan, Gromov-Witten Theory of P^1xP^1xP^1 http://arxiv.org/abs/1201.4414 (2012) This Article is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. GROMOV-WITTEN THEORY OF P1×P1×P1 DAGAN KARP AND DHRUV RANGANATHAN Abstract. We prove equivalences between the Gromov-Witten theo- ries of toric blowups of P1×P1×P1 and P3. In particular, we prove that the all genus, virtual dimension zero Gromov-Witten theory of the blowup of P3 at points precisely coincides with that of the blowup at points of P1 ×P1 ×P1, for non-exceptional classes. It follows that the all-genus stationary Gromov-Witten theory of P1 ×P1 ×P1 coincides with that of P3 in low degree. We also prove there exists a toric symmetry of the Gromov-Witten theory of P1×P1×P1 analogous to and intimately related to Cremona symmetry of P3. Enumerative applications are given. 1. Introduction While P3 and (P1)×3 are birational, it is too much to expect their Gromov- Witten (GW) theories coincide. Indeed, GW invariants are not preserved by birational transformation in general. Instead, one may hope to study special birational maps, such as crepant transformations or blowups. We prove the equivalence of the all-genus virtual dimension zero non- exceptional GW theories of four spaces, illustrated in the following diagram. Isomorphism GW(Xˆ) GW(X˜ˆ) Blowup Blowup Crepant GW(X) GW(X˜) Transformation arXiv:1201.4414v2 [math.AG] 18 Jun 2012 3 Here, X is the blowup of P at k points p1,...,pk, and Xˆ is the blowup of 1 ×3 X at six lines. Also, X˜ is the blowup of (P ) at k − 2 points p˜1,..., p˜k−2, and X˜ˆ is the blowup of X˜ at six lines. The equivalence of the all genus virtual dimension zero GW theories of X and Xˆ for nonexceptional classes was proved by Bryan-Karp in [4, Lemma 7]; in this work we complete the square. In more detail, let h ∈ H2(X; Z) denote the class of the proper transform 3 of a general line in P , let Ei denote the exceptional divisor above pi, and let ei be the class of a general line in Ei. Date: June 19, 2012. 1 We decorate classes on X˜ with ∼. Accordingly, let h˜j for 1 6 j 6 3 be the 1 ×3 classes of the proper transforms of the three lines in (P ) , and e˜i be the class of a general line in the exceptional divisor above the point p˜i. Theorem 1. As above, let X be the blowup of P3 at k points and let X˜ be P1 ×3 k the blowup of ( ) at k − 2 points. If β = dh − i=1 aiei ∈ A1(X) with ai 6= 0 for i > 4, then for any genus g, we have P ˜ X = X h ig,β h ig,β˜ , ˜ 3 ˜ ˜ k−2 ˜ where β = 1 dihi − i=1 a˜ie˜i and the coefficients of β and β are related by P P d˜1 = d − a2 − a3 d˜2 = d − a1 − a3 d˜3 = d − a1 − a2 a˜1 = a4 a˜2 = d − a1 − a2 − a3 a˜i = ai+2 for i > 3 . Remark 1. Note that the birational map X 99K X˜ is crepant. In general, GW invariants are not preserved under crepant transformations. This is the subject of the Crepant Transformation Conjecture; see [3, 7, 17]. However, Theorem 1 shows equality does indeed hold in the case considered here. Remark 2. Also note that we use the term nonexceptional rather strongly. Let π : Yˆ → Y be the blowup of the variety Y centered at Z ⊂ Y. We say β ∈ H2(Y; Z) is nonexceptional if any stable map to Y representing β has an image with empty set theoretic intersection with Z, and, moreover, any stable map to Yˆ representing βˆ = π!β has image disjoint from the exceptional divisor E → Z. We prove that the classes considered in Theorems 1 and 3 are nonexceptional in this strong sense. Now, let lij denote the line through pi and pj, and let Xˆ denote the blowup of X at the proper transform of the six lines lij, where 1 6 i < j 6 4. Let hˆ and eˆi denote the the proper transform of h and ei respectively. Theorem 2 ([4], Lemma 7). Let d, a1,...,ak ∈ Z be such that 2d = ai and ai 6= 0 for some i > 4. Then P ˆ X = X , h ig,β h ig,βˆ where β = dh − aiei and βˆ = dhˆ − aieˆi. P P Similarly, for (i, j) ∈ {1, 2} × {1, 2, 3}, let ˜lij ⊂ X˜ denote the line containing the point p˜i and representing one of the three line classes h˜j; see Figure 1 ˆ ˆ below. Let X˜ be the blowup of X˜ along the six lines ˜lij. Also, let h˜j and e˜ˆi denote the proper transforms of h˜j and e˜i respectively. 2 ˆ Figure 1. The six lines in X˜ blown up to obtain X˜ We prove that the all-genus virtual dimension zero GW theory of X˜ˆ is equivalent to that of X˜, in the nonexceptional case. ˆ Theorem 3. Let X˜ and X˜ be as above, and let d˜1, d˜2, d˜3, a˜1,..., a˜k ∈ Z be such that d˜j = a˜i and a˜i 6= 0 for some i > 2. Then for β˜ = d˜ih˜i − a˜Pje˜j, we haveP P P ˆ X˜ X˜ h i ˜ = h i ˆ , g,β g,β˜ ˆ ˆ where β˜ = d˜ih˜i − a˜je˜ˆj. P P ˆ In Section 3 we show that Xˆ and X˜ are isomorphic. Indeed, they are each isomorphic to a blowup of the permutohedral variety. Since GW invariants are functorial under isomorphism, this result, combined with those above, completes the square. We immediately point out two additional implications of this square of equivalences: enumerative calculations and toric symmetry. Remark 3. One may extract enumerative information directly from invari- ants of X and X˜, or one may relate invariants of X and X˜ to invariants of (the convex spaces) P3 and (P1)3. To accomplish the later, one may use the following result of Bryan-Leung [5], which generalizes a result of Gath- mann [12]. Let Y be a smooth algebraic variety and π : Yˆ → Y the blowup ! of Y at a point. Let β ∈ A1(Y) and βˆ = p (β). Then we have, ˆ p Y = Y , h ig,β h ig,βˆ −eˆ where eˆ is the class of a line in the exceptional locus, and p!(β) = [p⋆[β]PD]PD. 1 ×3 Example 4. How many rational curves in (P ) of class h1 +h2 +h3 pass through three general points? We compute (P1)×3 ˜ hp3i = h iX 0,h1+h2+h3 0,h1+h2+h3−e1−e3−e4 = X h i0,3h−e1−···−e6 6 P3 = hp i0,3h = 1. To illuminate, the first equality holds via Remark 3; we have blown up (P1)×3 along four points, and simply not used p˜2. The second equality follows from 3 Theorem 1. The third equality again follows from Remark 3. The final equal- 6 P3 ity holds as the invariant hp i0,3h counts the number of degree-3 rational curves in P3 through six general points. There is only one such curve, the rational normal curve. Additionally, the invariants on X˜ above satisfy a symmetry given by the following theorem. ˜ ˜ ˜ ˜ Theorem 5. Let X be as in theorem 1. Then if β = 16j63 djhj − 4 P i=1 a˜ie˜i, and {a3, a4} 6= {0}, we have P ˜ ˜ X = X h ig,β˜ h ig,β˜ ′ , ˜ ′ ˜ ′ ˜ 4 ′ where β = 16i<j63 djhj − i=1 a˜ie˜i has coefficients given by P P ˜ ′ ˜ ˜ d1 = d1 + d3 − a˜1 − a˜2 ˜ ′ ˜ ˜ d2 = d2 + d3 − a˜1 − a˜2 ˜ ′ ˜ d3 = d3 ′ ˜ a˜1 = d3 − a˜2 ′ ˜ a˜2 = d3 − a˜1 ′ a˜3 = a˜4 ′ a˜4 = a˜3. Acknowledgements. D.K. was partially supported by the Beckman Foun- dation and D.R. was partially supported by a Borrelli Fellowship during the preparation of this work. 2. Gromov-Witten theory We now briefly recall GW theory and fix notation. Let X be a smooth complex projective variety, and let β ∈ A1(X) be a curve class. We denote by Mg,n(X, β) the moduli stack of isomorphism classes of stable maps f : (C,p1,...,pn) → X, where C is an n-marked, possibly nodal genus g curve. This stack admits a vir virtual fundamental class [Mg(X, β)] of algebraic degree vdim(Mg,n(X, β)) = (dim X − 3)(1 − g)− KX · β + n. We denote by evi evaluation morphisms evi : Mg,n(X, β) → X defined by (f,C,p1,...,pn) 7→ f(pi). ∗ Let γ1,...,γn ∈ H (X) be a collection of cohomology classes. The genus- g class β Gromov-Witten invariant of X with insertions γi is defined by n X ∗ hγ1,...,γnig,β = evi γi. Z vir [Mg(X,β)] iY=1 4 For further details regarding the fundamentals of Gromov-Witten theory see, for example, the wonderful text [14].
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