ll ! I , WU1!“(HUIIWIWUIIIHUWNWWWI! HS l 9603 Was/701% LIBfi t Michiga State Unive .aty , This is to certify that the dissertation entitled REAL ASPECT OF THE MODULI SPACE OF STABLE MAPS OF GENUS ZERO CURVES presented by Seongchun Kwon has been accepted towards fulfillment of the requirements for the Ph.D degree in Mathematics \J Major Professor’s Signature August 17, 2003 Date MSU is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date clue. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 c:ICIRCJDateDue.965-p.15 REAL ASPECTS OF THE MODULI SPACE OF STABLE MAPS OF GENUS ZERO CURVES By Seongchun Kwon A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2003 ABSTRACT REAL ASPECTS OF THE MODULI SPACE OF STABLE MAPS OF GENUS ZERO CURVES By Seongchun Kwon We show that the moduli space of stable maps from a genus 0 curve into a nonsin- gular real convex projective variety having a real structure compatible with a complex conjugate involution 011 CI?”6 has a real structure. The real part of this moduli space consists of real maps having marked points on the real part of domain curves. This real part analysis enables us to relate the studies of real intersection (cycles with real enumerative problems. To my parents iii ACKNOWLEDGEMENTS I would like to express my gratitude and thanks to my advisor, Professor Selman Akbulut for suggesting this problem and continuous support. I thank Pro- fessor Sasha Voronov, Professor Sheldon Katz for getting algebraic geometry ques- tions and observing working ideas at, the starting point, Professor Rahul Pandhari- pande for reading and comments about informal details of construction of real part, real part analysis, Professor Pierre Deligne for reading my informal draft, sugges- tions,comments, answering my questions, Professor Frank Sottile for discussions and help on real algebraic geometry, Professor YongGeun Oh for his explanation about Fukaya-Oh-Ohta—Ono’s work, Gefry Barad, Professor Janos Kollar for helpful e-mail correspondences, Professor Michael Shapiro for reading and suggestions, Professor David Blair, Professor John McCarthy for help on English. I also thank I.A.S."s hospitality when I visited that place during spring 2002. Finally, I thank my parents for their encouragement, moral and financial support for a long period of time. iv TABLE OF CONTENTS 1. Introduction ............................................................... 1 2. Preliminaries .............................................................. 2 3. The moduli space of stable maps is a real moduli space ............... 9 3.1. Fulton-Pandharipande’s construction of the moduli space of stable maps MACH)", d) ............................................................. 11 3.2. Proof: The moduli space of stable maps is a real moduli space. . .. 20 4. Real part of the moduli space of stable maps and Projectivity ...... 27 4.1. Real part of the moduli space of stable maps ....................... 27 4.2. Projectivity of the real model ETJX, 3):“? ............................. 35 5. The Gromov-Witten invariant and real enumerative problems ...... 38 References .................................................................... 41 1 Introduction A Gromov-Witten invariant and its applications to enumerative problems in the com- plex world has been studied by many people. That invariant is defined on the moduli space of stable maps ( Definition in section 3 ). In this thesis, we investigate the real aspect of the moduli space MAX, 5) of stable maps from genus 0 curves when the target space is a convex (i.e. H1(CIP1, p*(TX)) = 0, for every ,u : (C1?1 —+ X, where TX is a tangent bundle) nonsingular projective real variety whose real structure cor- responds to the complex conjugation map on CW“. Here, a projective real variety is a projective variety having an anti-holomorphic involution. To search for the ways to use the above moduli space of stable maps TV!" (X, 5) in studying real enumerative problems, we have to see whether we can understand the moduli space Cliff-AX, 3) as a real projective variety or not. If the answer is positive, then we need to understand the nature of the real part ( Definition in sec 2), for example, whether each point in the real part of MAX, B) represents real maps or not. We will show the following: 0 (Section 3, 4.2) The moduli space of stable maps of genus zero curves WAX, /. ), where X satisfies the above conditions, is a real projective variety. 0 (Section 4.1) The real part of Mnflt’, ,8) consists of real maps having marked points on the real part of the domain curves. The real model (Definition in sec 2) of MAX, [3) has a Z-module Chow group fun- damental cycle. And the real part of MAX, ,3) has Z/2Zmodule ordinary homology fundamental cycle. So, it is natural to consider whether we can define real enu- merative invariants on the real model or the real part of ILL-Y, ,3) which count the number of real curves on the real model or the real part of iii-”(AZ ,3). Unfortunately, we cannot define nice enumerative invariants using fundamental cycles. The reason is explained in section 5. The possible way to use the real aspect of 37,,(X, .3) will be developing an efficient method to construct real cycles meeting transversally, maxi- mizing the number of intersection points of cycles in the real part of KTJX, 6). Its enumerative implication will concern how many real solutions we can have for the given enumerative problem, improving the minimum bound of the real solutions. But the technique to construct such real cycles is open. A Gromov-VVitten invariant in the real world with Quantum Schubert calculus has been widely studied by F. Sottile. See [Sotl], [Sot2], [Sot4], [Sot5]. 2 Preliminaries We begin with reviews of some standard notions and facts in real algebraic geometry. A more detailed exposition can be found in [Sil, I. sec.1,4]. Definition. Let X be a scheme over C. We will say that (X, s), or simply s, is a real structure on X if s is an involution on X such that the diagram X 1> X i l SpeC(C) 1; Spec(C) commutes, where j : C —+ C is the complex conjugation. We then call such a scheme X a. real scheme with a real structure 3. Remark 2.1 If X is a projective variety over C, then having a real structure is equivalent to having an anti-holomorphic involution on the set of complex points X(C). See [Si], p4, (1.4) Proposition]. Definition. Let X be a scheme over C. We will say that X has a real model if there exists a scheme XE over R such that X ”:3 XE XR C, where X3 x3; C is the fibre product of X R and Spec(C) over Spec(R). We will call XR a real model of X, and X a complexification of XE. Proposition 2.1 The category of quasi-projective or projective schemes ouer IR and that of quasi-projective or projective schemes over C, endowed with real structures are equivalent categories. More precisely, there exists a real structure (X, s) on a projective or quasi-projective scheme X over C if and only if there exists a real model X?R for X and an isomorphism (p : X —> XR XKC such that s = 99—1 0004;), where o is induced by complex conjugation in XE x3 C. For a fixed (X, s), tp and XK are unique up to real isomorphism. Proof. See [Si], p5] [:1 Definition. Let (X, s) be a real structure on a projective or quasi-projective scheme X over C. We will call the fixed points by the action s in the complex points X(C) the real part of X and denote it by X”. If a projective scheme Y is defined over R, then it consists of real and non-real closed points because the real number field R is not algebraically closed. A real model of the real projective scheme X is an algebraic geometric notion including non-real points also. But the real part of X(C) and the set. of real closed points in a scheme Y defined over R are differential geometric notions. If X(C) is real isomorphic to Y x13 C, then each point in the real part of X(C) uniquely corresponds to real points in Y, and vice versa. We will use notations ClP’k, RIP" to represent ProjC[;1:0, . . . , :rk], Pro jR[.r0, . . . ,xk], that is, projective k-spaces over C, R in algebraic geometry. Note that CP" can be considered as a k-dimensional complex projective space in the differential geometric sense. But RP'“, an algebraic variety containing non-real points, cannot be identified with a differential geometer’s real projective space. The set of real points in RIP” is identifiable to a differential geometer’s k-dimensional real projective space which we will denote it by RIP". Definition. Let (X, s), (Y,t) be real schemes. We will say the morphism f : X ——> Y is a real map if the morphism f commutes with real structures, i.e., f o s = to f. Such a morphism f obviously preserves the real parts, i.e., f (.X”) C Y”. If XR, YR are separated schemes of finite type over R, then giving a morphism f R : XER —> YR is equivalent to giving a morphism f : X —> Y which commutes with the real structures. See [Har, p107, 4.7. (c)]. We will call f3 a real model map of a real map f, f a complexified map of f3, the restriction map f“ : X” —> Y“ of f to the set of real points a real part map of f. Example 2.1 1. CPI“ is a real scheme having an anti-holomorphic involution given by a standard complex conjugation map. Then, CP'“ is isomorphic to RIP" xx C. We illustrate a non-real point in RIP”. Let OR be the standard open set {.170 7f 0}. Then, OR is isomorphic to SpecR[y], where y = xl/xo. Note that y2 + 1 is an irreducible polynomial in R[y]. Therefore, it generates a prime ideal and obviously corresponds to a non—real closed point. This non-real point splits into two complex points [1 : i], [1 : —i] corresponding to (y + i), (y — i) in a standard open set OR XR C isomorphic to SpecC[y] = Spec(lR[y] 89a C) in CPI. The set of these two points is preserved by the involution. In this special case of dimension 1, there is a set theoretic one-to- one and onto correspondence between CPI/ ~ and the scheme RP] , where ~ is the equivalence relation by the conjugation action, because every irreducible polynomial having degree higher than one has degree 2. Note that CPl/ ~ is diffeomorphic to a closed disk. More generally, real points in RIP" correspond to the points in the real 4 part of CPI“ which are fixed by the involution. Each non-real point in RP,“ splits into an even number of non-real complex points in CPI“ preserved by the complex conjugation action. 2. We may have more than one real structure on the same scheme. Not every real structure induces a real part. For example, Cli"1 has two non-isomorphic real structures. One is explained in 1, having a real part diffeomorphic to RIP”, with real model isomorphic to RIP]. The other is an anti-holomorphic map s([z : w]) = [—w : 23], having no fixed points, i.e. no real part, with real model isomorphic to the conic in RIP>2 given by the homogeneous equation x3 + if + x5 = 0. See [Har, p107, 4.7,(e)]. In general, if a smooth real scheme X has a real part, then the dimension of the real part is half of that of the original scheme. i.e., dim.3X(C) = dimRX". See [Sil, p8]. 3. The Deligne-Mumford moduli space TU" with it marked points, is a real moduli space whose anti-holomorphic involution is induced by the involution in example 1 as described in [G-M, 2.3], [F-Oh, sec.10], [Cey, sec.4.1]. More precisely, i. For non-singular curve; (CP1,a.1, . . .,a,,) +——> (CP1,(11,.. .,(‘1.,,) ii. For singular curve with two irreducible components; (CP;,6;(1,1,...,a.k) U (Cl?),(5’;b1,...,bl) H (CP;,S;a,,...,ak) U (CP:,5’;I.)I,...,l-)1), where CPL, CP}, are irreducible components after the normalization, (5, 6’ are gluing points, a,, bj are marked points, k + l = ii. iii. General cases are obvious from i, ii. We will prove this map defines an anti-holomorphic involution in section 3. In fact, this is an involution we get when we consider the C-scheme Deligne-Mumford moduli space as a complexification of the R-scheme Deligne-Mumford moduli space. That is, the real model of 17,, is an R-scheme Deligne-Mumford moduli space T1: with 72 marked points. Not every universal family of curves 3,, on 37,, is real. For example, any rational curve with 3 marked points can become a universal family over .113. But that can be considered as a real universal family of curves only when all 3 marked points are on the real part of CP’. To analyze the real part of fin, we need a real universal family of curves. The real universal family of curves can be constructed by complexification of the R-scheme universal family of curves over the R-scheme Deligne-Mumford moduli space Hf. Obviously, the one point moduli space H3 can be represented by a rational curve (CPI, a1, a2, (13) with 3 marked points a,- in the real part. Each general point x(;£ a,) in U3 1: (CP’, al, a2, a3) can be understood as a general point in H4 representing a rational curve with marked points at al, a2, a3, x. Thus, non—singular curves represented by points in the real parts of H4 are real curves with 4 real marked points. When these curves degenerate, they make singular curves having real irreducible components with real marked points and real gluing points. If two points on the real part of the rational curve collide, then the colliding place becomes a gluing point with the other new real irreducible component. And then, the collided points split into two points in the real part of the new irreducible component. Since the construction of Deligne—Mumford moduli space is inductive using real isomorphisms Un_1 E’ I17", we see that points in the real part of 717", n 2 3, are represented by rational curves with real marked points or singular curves having real irreducible rational components with real marked and real gluing points. See [G-M, sec2.3]. Recall there is a 1-1 correspondence between isomorphism classes of locally free sheaves of rank n on the scheme X, and isomorphism classes of vector bundles of rank it over X. We won’t distinguish the words between a ‘locally free sheaf’ and a ‘vector bundle’. See [Har, p129, 5.18(d)]. Note that a real structure 5 on X induces a canonical morphism on the structure sheaf 0x by F(U7 OX) —) I‘(3(lj)701\') f H j0f08:=f"’3 which is an isomorphism of rings, where U is any Open set in X. Let U be an affine open set in X and .C a locally free sheaf. Then, £(s(U)) is an Ox(s(U))-module. We make £(s(U)) an OX(U)-module by changing exterior multiplication, 0x01) >< £(«S(U)) -> C(b'(U)) (f, v) i——) (j o f o s)v leaving the underlying additive group structure as it was. We call the locally free sheaf £3 of (OX-modules defined in this way the conjugate vector bundle with respect to the real structure 3 on (X, s). For example, if £ is a sheaf of functions with values in C’, we may describe £3 by £‘°(U) : {j 0 ho slh E £(s(U))} We call the vector bundle V over the real scheme (X, s) a s-real bundle if its conjugate bundle Vs is identical to the bundle V. Here, ‘identical’ means exactly the same, not meaning isomorphic. The line bundle from the structure sheaf OX on the real scheme (X, s) is a trivial example of a real vector bundle. Remark 2.2 Let D = Zn,D,~ be a Weil divisor on a real scheme (X, 3). Let DS be a conjugate Weil divisor En,s(D,~). If we consider a Cartier divisor {(0}, f,)} associated to the Weil divisor D, then its conjugate Cartier divisor, the Cartier divisor associated to the conjugate Weil divisor D“, can be written as {(U,, ff)}. Hence, if 0(D) is the invertible sheaf associated to D, then its conjugate line bundle comes from its conjugate Weil divisor, i.e. (O(D))“‘ = 0(D3). Conversely, if .C is an invertible sheaf on X and D(£) is the associated Weil divisor, then the associated Weil divisor for the conjugate line bundle .63 is the conjugate Weil divisor of D(£), i.e., D(£“’) = (D(£))3. Consequently, the line bundle is real if and only if its associated Weil divisor is fixed 7 by an involution s. More generally, the vector bundle V on X is real only when there exists a locally free sheaf l? on XLR whose complexification becomes V. See [Si], p6, (1.8) Lemma]. Example 2.2 1. Line bundles on CP", RIP" are classified by their degree. That is, any invertible sheaf on CP", RPk is isomorphic to (9(1) for some I E Z. See [Har, p145]. But the restrictions of same degree line bundles on 1R1?" to the real points, so line bundles on the differential geometer’s real projective space RIP", are not necessarily isomorphic. Let s be a real structure from the complex conjugation map on CP’. Then, the Weil divisors [2' : 1] + [—i : 1] and [1 : 1] + [—1 : 1] define degree 2 s-rea] line bundles, say L1, L2 respectively. The natural holomorphic section 31 to L1 induced from the associated Cartier divisors( see [Grif-H, p135] ) vanishes at. [i : 1] and [—i : 1] which are not in the real part of C11”. Thus, it induces a trivial line bundle on RIP”. But the s-real line bundle L2 induces a nontrivial line bundle on RIP”. If we restrict the s-real line bundles to the upper-hemisphere so that the fibers along the boundary come from the real parts of L1 and L2, then these give an example of line bundles whose Chern classes are the same after the complex double, i.e. in this case, line bundles L1, L2 on CP’, but the real line bundles along the boundary are not isomorphic. The invariant for line bundles on the upper-hemisphere is called a relative Chern class. 2. Not every degree’s line bundle on Cl?”c allows a real line bundle. Let s be a real structure on C11”1 from the antiholomorphic involution [z : w] i—> [—w : :3], which doesn’t have any fixed point. Then, this real structure doesn’t have any odd degree real line bundles O(2r+1) because there is a 1—1 correspondence between Weil divisors, and invertible sheaves ( see [Har, p144] ) and none of the odd degree’s Weil divisors can be fixed. Remark 2.2 leads us to the conclusion. 3 The moduli space of stable maps is a real moduli space The moduli space 37,,(X; '3) of stable maps (f, C, .271, . . . ,.1:,,) from a germs zero curve with n-marked points consists of the equivalent classes of stable maps (f, C, x1, . . . ,x,,) satisfying the following conditions by its definition; (1) f...( [C]) represents the homology class ,13 in H2(X; Z); (2) The arithmetic genus of domain curves having “ii-marked points is zero ; (3) (stability condition) If the domain curve C has some irreducible components C? such that f..([C,’-)]) = 0, then each of these components, C9, contain at least 3 special points(marked or gluing points); (4) Two stable curves (f, C, x], . . . ,1:,,). (f’, C', x’l, . . . ,.r:,) are equivalent. if there exists an isomorphism o; C —> C’ such that f’ o o = f and o(.r,~) = x], i = 1,. . .,n. Let (CP’,s), (X, t) be real structures. Then, it is natural to be concerned whether the set theoretic involution (f, CP’, x1, . . . ,xn) i—> (tofos, CPI, s(.r1), . . . , s(x,,)) de- fines an anti-holomorphic involution on 37,,(X; ,3). We will consider the real structure coming from the complex conjugation map on CPk and real projective varieties X related to this real structure and show the above involution is an anti-holomorphic involution on 71/7,,(X; [3). At the end, we will see this result doesn’t. always hold for any real structures on a domain and a target space. We will follow Fulton-Pant]haripande's construction in [F-P]. The moduli space of stable maps of genus zero curves was constructed by gluing the quotient of projective varieties which are the universal space of an h-rigid stable family of degree (1 maps (See a section 3.1 for the definition). The strategy for showing the moduli space of stable maps is a real moduli space. is showing each of the ingredients they used are real. The universal space for an h—rigid stable family was constructed by using a certain locus of the Deligne-Mumford moduli space and a universal curve on it. Their construction is not dependent on the chosen universal curve model. However, we need a real universal curve model for our proof. The existence of a real universal curve was explained in Example 2.1,3. Lemma 3.1 The Deligne-Mumford moduli space Hm is a real moduli space with a real structure induced by a complex conjugation map on CP’. Proof. method 1: (simplest) The Deligne-h‘lumford moduli space is originally defined over Z. So, it is defined over any field. The C-scheme Deligne—Mumford moduli space can be obtained by a scalar extension from the lit-scheme Deligne-Mumford moduli space. That is, the C-scheme Deligne-Mumford moduli space is a complexification of the R-scheme Deligne-Mumford moduli space 3715,. So, Lemma 3.1 is proved by Proposition 2.1. method 2: (geometric) We consider the involution defined in Example 2.1 3. The map we defined is an antiholomorphic involution because the image curve’s marked and gluing points are induced by the complex conjugation map on that curve and the splitting of a tangent space at (C, a1, . . . , am) is; T(0,0,1, . . . , unfit" g H1(C,7Z;(—a1... — and) GB EB‘sésmg(C)T§ ‘39 T: E“ €90,,:irreducszieHl(Cm Teal-'01 - - - — 0a)) G @sesing(C)Ts' 8’ T: 9: $00,,,,,,duc,bleH0(Ca,721((11 + + aa) 83 won)“ GBEBSESingijS’ (8) T;’, by Serre’s duality. E] .4 Remark 3.1 Araujo - Kollar constructed the moduli space of stable maps on any Noetherian scheme in [A-K, sec.10]. However, the relation between an lit-scheme version’s moduli space of stable maps and a C-scheme version’s moduli space of stable maps is different from that of R-, C-scheme Deligne-Mumford moduli space. That is, a C-scheme version’s moduli space of stable maps is not the complexification of an R-scheme version’s moduli space of stable maps. A counterexample showing that. the real model of the moduli space of stable maps and an lit-scheme version’s moduli 10 space of stable maps are different is given by, z i—> 7:2 and z 0—) —22. These maps are different in R-scheme version’s moduli space of stable maps but they are equivalent in the real mode] of the moduli space of stable maps .1/IO(CP’,2) by an isomorphism between the domain curves defined by multiplication by i. 3.1 Fulton-Pandharipande’s construction of the moduli space of stable maps MACH”, d) Fulton-Pandharipande constructed the moduli space 11—1,,(C1Pk, d) for k > 0, d > O and (n, k, d) 75 (O, 1, 1). Other cases, 21—1,,(ClP’0, 0), 717,,(CPk, O), 370(CIP”, 1) are isomorphic to 11—1", 117,, X CP”, S pec(C) respectively. [I] Construction of the universal space 37,,(C1P’k, d, h) for the l—i-rigid stable family of curves We call the correspondence between irreducible components of the curve C and the degree of the restriction of the line bundle .C to each component of C as the multidegree of .C on the curve C. We will say bundles L, V on C satisfy equal mutidcgree condition if their degrees on each component are the same. Definition. [F-P, 3.2] Let crk = em), where H* 2 H0(ce’c,om,k(1)). Let h : (ho, . . .,hk) be an ordered hyperplane basis of H“. A h-rigid stable family of degree (1 maps from n-pointed, genus 0 curves to Cl?”C consists of the data (7T 3 C "t 5» {Piligigm {(Ii.j}05igk.igjgd-/ll. where (i) (it : C —> S, {pi},/2.) is a stable family of degree (1 maps from n-pointed, genus 0 curves to CPk, where u; C —+ CP"; (ii) (7r : C —> S,{1),}1sign,{(5,303,9’199) is a flat, projective family of n. + (1(k +1)- pointed, genus 0, Deligne-.\'Iun’1for(_l stable curves with sections {p,-} and {(11.1}? (iii)(Transversality condition) For 0 S i g k, there. is an equality of Wei] divisors 11 MUM.) 2 (12,1 + qi,2 + - - - + (1232:- Remark 3.2 1. An l—z-rigid stable family is a special kind of flat family of degree (1 maps from n-pointed genus O to CP" such that the image of each fibre curve inter- sects each chosen hyperplane basis (ho, . . . , hk) of P(V) transversally at unmarked, nonsingular points. 2. The condition (iii) implies the last d(k + 1)-marked points {qij} are from the hyperplane intersection divisors. Fulton-Pandharipande added those ordered hyper- plane intersection marked points to relate the geometry of the moduli space of stable maps of genus zero with that of Deligne-Mumford moduli space. 3. Note that the condition (iii) combined with (i) implies that the number of marked points from each set of {gm}, i = 0, . . .,k, on each irreducible component in each fibre is exactly the same as the degree of the map on each component. That implies k + 1 line bundles on C constructed by using Weil divisors (11,1 + . .. + g“, from the last d(k + 1) marked points satisfy the equal multi-degree condition. There is a universal locus B in Deligne-h-lumford moduli space fim, in : n+d(k+ 1) that every h—rigid stable family in (ii) factors through. But the map’s information we can get from the points in B is limited to the hyperplane intersection points. To distinguish the h—stable maps sharing the same hyperplane intersection points, Fulton-Pandharipande constructed a k-dimensional C*-fibration on B by using the k + 1 Weil divisors (1231 + + gut. The following notion of H—balanced is satisfied by the sublocus B and enables them to construct the desired fibre bundle, which is a universal space for the h—rigid stable family of maps. Notation. We will denote the line bundle 017m ((1,,1 + (1,32 + . . . + qnd) on D", by 7-1, i = 0,. . . , k. Definition. [F—P, 3.3] Let. .zTI—m be the Deligne-Mumford moduli space of genus 12 0, m-pointed curves. Let it : U m —> 111m be the universal curve with m-sections {Pillsisn and {q,,j}03.,gk,1sjsd. For any morphism 7 : X —~) Mm, consider the fiber product: , —r 7 —r' A X Hm D m _) U m i 77X i W I 7 — (X _> .A' I f” The morphism '7 : X ——> ll—Im is "H — balanced if (i) for 1 g i g k, 7r,\',.'7*(?l,- 8) H61) is locally free; (ii) for 1 S i S k, the canonical map axrrxfi’nti <8) 7-161) ——2 5"“(7-l, 8) Hg’) is an isomorphism. The condition (ii) implies that X goes to the locus in Wm satisfying the equal mutidegree condition for any pair of line bundles (’Hi, H0), i = 1, . . . , k on each fibre of the universal curve Um [7“). The reason is direct image sheaves may change the rank of the sheaves. If that happens, then the pull back of the bundle iri-rrxfl*(7-l,- @7151) in (ii) has different rank, preventing it from becoming isomorphic to 7*(H, <8) H61). Examples showing the rank changes of direct image sheaves are the following : Let it : CP’ -—> SpecC. Then, ”*(Ocel (1)) = HOWE”. Ocel (1)) E” C 69 C ”*(Ocell = H°(C1P”,0ce) ’5 C ”*(Ocelf—lll = H°(C1P”. cred—1)) g 0 and 7r*7r,((9mt (1)), 7r*7r,(0,cpi), 7r*7r,((’)C?1(—1)) are trivial bundles of rank 2, 1, 0 on C]?1 respectively. We can calculate direct image sheaves for the reducible curve cases by using a short exact sequence of locally free sheaves related to a normalization, by noticing that a genus zero curve is a tree, which implies the number of gluing 13 points is one less than the number of connected components, and taking a long exact sheaf cohomology sequence induced from that. What we can see is the rank of the line bundle is preserved by 7r*7r, only when the line bundle. is trivial on the fibre. Hence, the image of X by a morphism 7 sits inside of a certain locus in ‘17", on which the 91,8) ”HO— 1 are trivial line bundles on each geometric fibre, equivalently, the locus satisfying equal multi degree conditions for any pair of (White), i = 1,. . .,k. The universal sublocus B in Tim for the flat families in (ii) of the definition of the l—i-rigid stable family is the largest sublocus satisfying equal multi degree conditions for any pair of (71,310). Then, B is closed by an upper semicontinuity property [Har, p288]. By vanishing of higher direct image sheaf 72in... for i 2 1 and the Cohomology and Base Change Theorem [Har, p290], the. direct image sheaf of the line bundle H,- ® 713‘ on the universal curve Um becomes a well—defined line bundle on the locus B in 217,". The subscheme B itself is ’H-balanced by an inclusion map to W17", because the natural morphism 7r;(7r3,(”H,- @751)» 8301 C —> (’H, @7161): (80: C is surjective for all x E 7r’1(B) by noting ng(7r3,(7-l,- ® 7161)); (8103 C is isomorphic to the global section sheaf of ’H, ® ”H51 on the fibre of ng’(7rg(.r)). Lemma 3.2 The universal closed sublocus B in 211 m is a real projective variety. proof. Since B satisfies the equal nmltidegree condition, for any i and a chosen irreducible. component in the fibre over b E B, the number of marked points from ((1,, ills jgd is the same. The closed sublocus B in the projective variety .717", is invari- ant under the antiholomorphic involution described in 3 in Example 2.1 because the involution preserves the number of marked points from {qt jllg Ed on each irreducible component in any pair of conjugate curves. The Lemma follows from Lemma 3.1. E] Before we. do a fibration over B, we see how the fibration can compensate the. missing data with an example. Recall the following standard facts. Lemma 3.3 [Har, p150] Let C be a scheme over C. 14 If .C is an invertible sheaf on C, and if 80,...,Sk E H°(C,£) are global sections which generate C, then there e;1:ists a unique morphism «,9 : C —> Cll’k such that E“ 99*(0C?k(1)) and s,- : gs*(uy) under this isomorphism. Lemma 3.4 [Har, p157] Let C be a nonsingular projective variety over C. Let Do be a divisor on C and let I: E” C(DO) be the corresponding invertible sheaf. Then, (a) Every eflective divisor linearly equivalent to D0 is (8)0 for some 3 E H0(C,£), where (s)0 denotes the divisor of zeros of s. (b) Two sections s. s' E H0(C, .C) have the same divisor of zeros if and only if there is a A E C* such that s’ 2 As. Example 3.1 Let’s consider the geometric fibre on a geometric point in B, i.e., 7r}; : (ClP’l; {pi}, {q.,:,J-}) —+ SpecC ’5 b E B. We will use each set. {(11J}]Sjsd,0 S i g k from the last (1(k + 1)-marked points as a Weil divisor, so, effective Cartier, (12,1 + . . . + (125d- To use Lemma 3.3, we have to use. one line bundle and select 13+ 1 global sections 3,, telling the actual morphism to ClP’k, satisfying the condition (iii) in the definition of a h—rigid stable family of curves, i.e., vanishing at {gm-Lg“, for each i = 0, 1, . . . , k. Let’s consider a line bundle 0c?1((10,1 + . . .+q0,d) although we may consider any other line bundle OC?1((12',1 + + (1,0,). Note that all same degree effective divisors are linearly equivalent since line bundles over ClP’k are classified by their degrees. Hence, we can choose k + 1 global sections 3,- E H0(CP1,OC?1(210,1 + + (1%)) satisfying the requirement by Lemma 3.4 (a). On the other hand, the linear system on CH”1 generated by 3,, i = 0,. . .,k, , has no base point because the (11.1 are distinct points in CH”1 . Thus, we can use Lemma 3.3. Now, we describe the morphism to Cl?" decided by the chosen global sections 3,, i = 0,...,k. Let’s denote SO 2 {p E Cll’llso(p) doesn‘t vanish } and U = {uro 75 O} C ClP’k'. Then, the actual morphism restricted to So —> U comes from the ring homomorphism C[::1, . . . , ck] —> C(SO, (950) by sending 21‘ H 53/80, (1) and making it C-linear, where 2:,- = w,/u.v0, i = 1,. . . ,k. Since the set 50 is ClP’1\ finite points, the above restriction map is uniquely extended to the whole space Cll’l. Observe that the morphism in Lemma 3.3, i.e., the morphism (1), is dependent on the actual choice of s,- E H0(ClP’1, O,g;;1((10,1 + - - ° + q0,d)), but up to ratio of .s,/.90, i = 1,...,k. More precisely, {A0s0,...,)\ksk}, {Ag.s-0,...,/\’,.s,,}, /\,~,/\; E C" induce the same morphism if and only if /\,/)\0 : Ag/XO, i = 1,. . . , k. Constructing a space recording all possible ratios A,s,-/)\0so, A,, A0 E C“ is our goal. Then, there will be a one-to-one correspondence between points in the constructed universal space over SpccC ’5 b and maps whose hyperplane intersection points are {qi‘j}13jgd,i= 0, . . . , k. Let s,, i = 0,. . . , k be the chosen global sections. Observe that invertible sheaves 7i,- ® 7151 are generated by any r,(sO/s,-), r,- E C‘ and the coefficients r,-, which are degree 0 polynomials, can be considered as elements in the H0(CIP’1,7-l,- ® H61). Therefore, all possible ratios /\,-s,-//\0so, A,, A0 E C“ can be recorded by [HO(C1P’1,’H1® H61) \ 0] x X [H0(CP1,Hk (8 H61) \ 0] E’ C’ X x C‘, where H,- : 03?; ((11.1 + . . . + gm). Let’s summarize the geometric procedure of the above construction. Let 7r : (ClP’l, {1),}, {q,-.J-}) ——> SpecC be a geometric fibre on the geometric point b in B and so, . . . , sk be global sections in H 0(ClP’l, ’HO), whose zeros generate effective divisors (Ii,1+...+(1,',d, i = 0,. . . , k. We constructed bundles ’H, = 03:31 (21,-‘1 +. . .+(1,-,d) using the last d(k + 1) marked points and considered the tensor bundles H,- ® "HO— 1 on CT“. Then, we considered the direct image sheaves 7r13,(7-l, <8) H51) ’5 CC, deleted a zero element from each i z 1,. . . ,k because of Lemma 3.4 (b), denoting them by 16 1?, and constructml a kerlimensional C*-bundle Yb :: Y,” x , ,. x 1ij on SpecC E“ I), q k “b '2; , 1 , 1 _r CIP’ <— 1 xi; CIP’ —> CIP> L) BL 0,", 7r}? .L i 7T3; i 7TB Yb 2: SpecC ”-3 b <—> B C .7le where BUG”, is the restriction of the universal curve L7", over .l—Im to B. The fibres of any elements in the k—dimensional C*-bun(lle Yb are naturally equipped with k+1 sections in H 0(Yb XCCP1,7b*(H0)), representing pull-back divisors "7b*((1i,1 + . . . + (1”,). By Lemma 3.3, there is a morphism ,u. to Cl?“ whose restriction to each fibre over Yb is similar to the morphism described in (1). In fact, that morphism is given by z,- +——> r,-(y) -"_,w”*(s,-)/e7b*(s0), where y E Yb, 1‘,- is a C‘-valued function on Yb which may be understood as an i—th projection map from Yb to 1;” = H°( B 7TB,\‘ /"Yi 1",- VVe observe the following: LTD-l,- ® H51) ”:‘1 'T/*7r;37r3,(7-l,- (8) H51) by the second condition in ’H—balanced I 2 _1 ”dfléfliflBJ’Hz‘ ‘8’ Ho l IIZ 2. yfirB,(’H,- (8) H51) has a tautological section because of the definition of 1"}. 3. The pull-back of the tautological section to ngngiy; nB,(’H,®’Hg 1) gives a globally non-vanishing section. 2 and 3 imply 71%,- ® ”HO— 1) is a trivial line bundle with tautological non-vanishing sections 1“,, i = 1,...,k which are constant along the fibres of Y. We can treat those sections r,- as functions from Y to C“. Now, we got the desired canonical induced sections r,"y*(s,-) E H°(Y x3 BUm,'3"*(H0)), i = 0,. . .,k, where r0 = 1, and r,- : Y —> C*. Lemma 3.3 gives a morphism from Y x3 BU", to ClP’k such that [131]”): 7'1”)”(81'), l = 0, . . .,k, and [t*(OC?A-(1)) g ’7,"(H0). All we have explained is the following Proposition. Proposition 3.1 [F-P] The moduli space of h-rigid stable family of degree d maps from n-pointed, genus 0 curves to ClP’k is a fine moduli space EACH”, d,h) which is a nonsingular projective variety. 18 [II] Quotients and gluing; The moduli space of stable maps of genus zero .l—I,,(ClP’k,d) was constructed by gluing quotients of the moduli spaces of h—rigid stable family of maps W,,(Cll’k, d. h), where h is any basis of H * = H 0(ClP’k , OClP’k (1)). We need to consider the followings: 1. Is there any ordered basis h = (11.0,. . .,hk) in H” for a given n-marked, genus 0, degree (1 curve to ClP’k, such that the curve intersects with any chosen hyperplane basis It,- transversally at unmarked, nonsingular points? That is, can we get. enough gluing pieces from the HACK)", d, h)? 2. The last d(k+1)-marked points {qi,j}ISde, i = 0, . . . , k played a role as hyperplane intersection divisors q“ + . . . + (1,4,, i = 0, . . . , k. How can we forget orders of points in each set {thy-hag, i : 0,. .. ,k ‘? 3. How can we glue quotients of moduli spaces 11—1,,(C1P’k, d,h) for various choices of basis h of H *'? The answers are the following: 1. Bertini’s theorem tells us that most hyperplanes in H * intersect with the given curve transversally at nonsingular points. So, we can always find the ordered basis h satisfying the conditions. 2. We make the product of the symmetric group G 2 GS x . . . x G; act on the moduli space (7r : L! —> ET,,(ClP°k,d,h),{1),},{(1,,j}OS,-Sk,1SJ-Sd,p), where Cf, acts on the set {Quallsjsd by permuting the orders. Since the finite group G acts on the projective variety A—[n(ClPk,d,h), its quotient. HACK” , d, h)/ G is also a projective variety. 3. Let. h, h’ be different choices of basis of H *. There are G-invariant open subloci W02, h’) , MUi’, h) in HH(CPk, d. h) , .TT,,(ClP°k, (1,?) respectively, consisting ofcurves intersecting with all hyperplane basis ho, . . . , hk, hf), . . . , I22. transversally at nonsingu- lar unmarked points. Clearly, il—[(h,h’) and TlT(h’,h) are isomorphic. And Fulton- 19 Pandharipande showed fi(h,h’)/G and 37(h’,h)/G are also isomorphic in [F-P] Proposition 4. 3.2 Proof: The moduli space of stable maps is a real moduli space First, we show that the ingredients used in Section 3.1 [I] are real with respect to the antiholomorphic involution induced by a. complex conjugation map on C1“, ClP’k. Lemma 3.5 The fine moduli space 111,,(C1P’k, d, h) is a real projective variety whose real structure is induced by COfllplCIIJ conjugation maps on CH", CH”, where h 2 (ho, . . . , hk) is a real ordered hyperplane basis of H0(ClP’k, OCPk(1))' proof. By Lemma 3.2, B is a real projective variety. Let in; : BU", —+ B be the real universal curve with m real sections {1),}199, and {q,,j}05,_<_k,15jgd from B to Bl—Tm. Then, the Weil divisors, (1m +. . .+q,~,d, i = 0, . . . , k and (12,1 +. . .+q,,d—q0,1 —. . .—q0,d, i = 1,. . . ,k' are all invariant under the anti—holomorphic involution on BUm. That implies the associated line bundles ”Hi, ’H, ® ”H; l are all real line bundles by Remark 2.2. Equivalently, there is an anti-holomorphic bundle involution on each bundle. Let r, e, 7‘,’ denote anti-holomorphic involutions on B, ijm, H1®HJ 1 respectively and ”is be a natural projection map from the line bundles ”H,- ® 715‘ on BU," to BUm. 20 | \h BU m —> BU", Wei i713 ~I ,,_®,,_1| , “have H, Ms, , z 0 7734(0)) 1 0 r§‘(C.a)) ”'3 i i W's 1N {pint—31(1)) N l (Clix, = C5 —) Cflbj 2 Cl? We’ll show the line bundles 7TB,(’H,- <8) 710-1) have natural anti-holomorphic bundle involutions induced from the anti-holomorphic bundle involutions on the H,- ® 7‘10— 1. Since n3,(7-l.- 68> H61) is a bundle over the real scheme B, it is enough to show that there is a natural anti-holomorphic involution between fibres over b and r(b). Let‘s see the bundle map iflflgl (Cb) restricted to the pointed curve Cb E «g1(b) and its pointed conjugate curve CT“) E ugl(r(b)), where b E B represents a nonsingular pointed curve isomorphic to C11“. Then, Cm) is isomorphic to C11"1 with conjugate marked points. For notational convenience, we denote both Cb, Cm) as ClP’l. Since the divisor Di,b E (127.1 (b) + . . . + q,,d(b) — qoj (b) — . . . — q0,d(b) has degree zero on Cll’l, it is a principal divisor. Let Di,b be defined by f, - fo‘l E H 0(ClP’l, IC“), where IC‘ consists of invertible elements in the sheaf of total quotient rings of Owl. Then, Orczs1(D,~,b) is 21 globally generated by f0 - ffl. The divisor me) on the conjugate curve is defined by fB-fi—l E HO(ClP1,lC*) because Dig“) is 6,,1(b)+. . .+(j,~,d(b)—(]0j(b)—. . .—(]0,d(b), where f,- is a conjugate polynomial whose coefficients are complex conjugates of the f,-. Then, 0m}: (Dmm) is globally generated by fo- f—fl. So, the restriction of the globally defined anti-holomorphic bundle involution 7‘,’ on ’H, <8) 713‘ to the map between O€P1(D,j,) and OCP1(D,-,T(b)) is the map sending f0 - ffl to f0 - f—fl. We can describe the similar situation when b E B represents a pointed singular curve with a little more work by using the sheaf exact sequence of a normalization. The canonical anti-holomorphic bundle involution r; on 7TB,(7-l,- <8) ”Hg 1) is induced from ”fi’. Observe the restriction map Ti'jb : n3,(7'l,- <8) 7151”,, —-> NBA/Hi <8) HJIMTM) can be considered as a complex conjugation map on the induced local charts because 7r3..(7-£.,- ® Ho’l) b ”:‘2’ H0(CP1,O.CF.1(D,,,,)) ’5 C and the bundle map r,’|7r2_1(cb) goes a - f0 - ff1 r—> d - f0 . ff] for any a E C, where a denotes a complex conjugate of 02 E C. This shows T,’ is an anti-holomorphic bundle involution on r3,(’H,~ 63) 713‘). Let 1‘}, i = 1, . . . . k. be the C*—bundle coming from n3,(?'l,- 8) H51) by removing a zero section. The restriction maps of r,’ to 1'}, i : 1,. . .,k are anti-holomorphic C*-bundle involutions. The k-dimensional C*—bundle Y E Y1 x3 x3 Yk has an induced anti-holomorphic bundle involution. That means fin(ClP’k,d,h) E Y is a quasi-projective real variety. Since the moduli space of l—z-stable degree d, n-pointed curves WACIP" , d, h) is a fine moduli space, we want to show 117,,(C1P’k,d,h) is equipped with a real universal curve and a real projective morphism from the universal curve over 37,,(le’k, 21, T2.) to err. Note that. Y has a universal family Y x3 BU," induced from the real universal family BU", ——> B C Em and Y x 3 BC", has a. natural morphism to ClP’k as explained in section 3.1 [I]. We will show Y x 3 BB", is real and u is a real morphism to ClP’k. It is easy to see Y x B BU", is real. Note that. Y, B, BU", are real varieties. Since 22 a question is local, we may consider Y, B, BU", as Spec(y_n.g @R C), Spec(BR <83 C), Spec(BURm @341 C) respectively. Let’s denote 32R (83:, C := 32, BR (8319; C := B, BUR", (83 C :2 BUm, real models Sp8(5(ng) :2 YR, Spec(BR) :2 BR, Sz)ec(BU3m) := BUS 3’ (EB BU"; g [ya (33],, (Be Gila Cl] (33,, @R C (BURm ®R C) (yR ®BR BURm) ®R C HZ This means Y x3 BU", 2 (YR x BR BUi) xR cc. . . . . ——IR Fmally, we see that there 1s a canonical morphism uIR from YIR x BR BU m to RI?" by the similar construction we saw in the section 3.1 [I]. Therefore, we may consider the morphism u as a complexification of uk. So, the Lemma follows. [3 We are ready to prove the main Theorem in this section. We will consider the similar questions written in section 3.1 [II] in a real setting. Theorem 3.1 The moduli space of stable maps of genus zero Hn(ClPk,d) is a real moduli space whose real structure is induced from anti-holomorphic involutions by complex conjugations on Cll’l, ClP’k. proof. Recall that Fulton-Pandharipande’s construction was about the moduli space MACH”, d) for k > 0, d > 0 and (n, k, d) yé (0,1,1). Other cases, HACK“), O), MACH”, 0), 170(C1P’1, I) are isomorphic to fin, 717,, x ClP’k, S pec(C) respectively and so, they are obviously real moduli spaces. We showed L7,,(le’k, d, h) is a real fine moduli space, where h is a real ordered basis of H * = H 0(C1P’k , OCPk(1))' We have to consider the following questions : 1. Is there any real ordered basis h 2 (ho, . . . , hk) in H * for a given n-marked, genus 0, degree (1 curve to ClF’k such that the given curve intersects with any chosen hyper- plane basis h, transversally at unmarked, nonsingular points? 23 2. Does the product of the symmetric group G action on the moduli space ill—1,,(le’lIt , d, h) commute with the anti-holomorphic involution on EACH”, (1, h) so that the quotient space EACH” , d, h) / G has an anti-holomorphic involution, (i.e. becomes a real variety)? 3. Does the gluing commute with an anti-holomorphic involution so that the anti- holomorphic involution on each MACH”, d, h) / G extends to the whole moduli space 17,,(crk, d)? The answers are the following: 1. As F. Sottile [Sot3] pointed out, real points are Zariski dense in ClP’k. That implies we can always find a real ordered basis satisfying the transversality condition. 2. The symmetric group G action on 17,,(C1P’k,d,h) to forget the orders of the marked points {gig} in each of the last k + 1 sets commutes with the anti-holomorphic invo- lution. (Cb, {Pi}: {(11.1}, Nb) it (Cb: {pilv {Qua-(1')}. ltb) I..€ \L 7" i TI 0' (CT(b)3 {iii}: {aidia fir(b)) _‘> (CT(b)7 {pi}, {qi,0(j)}3 fir(b)) ,where r’, T denote the involution on Hn(ClPk,d,h) described in the proof of Lemma 3.5, on B respectively, and o is an element in G. Note that the description in the above diagram is about up to isomorphism according to the equivalence relation in the moduli space 717,,(C1P’k, (1, 71) rather than about the actual model (Cb, {1),}, {q,,(,(j)}, uh). But there is no problem. For example, if (Cb, {pi}, {(1,300)}, uh) is isomorphic to (Cg, {pi}, {qu}, pbr), then (C70,), {13,-}, {q,,a(j)},flr(b)) is isomorphic to (CW),{13;},{gffij},;7,(b2)). More pre- cisely, if the linear fractional transformation (az + b) / (c2, + (1), ad — be gé 0, gives an equivalence relation between (Cb, {1),}, {qwm}, [11,) and (C52, {1):}, {Qijlv It”), then the 24 linear fractional transformation ((13 + (3)/(52 + d) gives an equivalence relation be- tween (Cab), {13,-}, {(‘1',,,,(j)},fl7(b)) and (6.452), {132}, {fight-Iraq), where a, b, ad are the complex conjugates of a, b, c, d if Cb is nonsingular. 3. It is easy to see that the isomorphism which is a gluing map between Em, l—LI) / G and Kflh’, h) /G commutes with the anti-holomorphic involutions on TV!" (ClP’k, d, h) / G and on Mn(ClP’k,d, 717) /G from the proof of [F-P] Proposition 4. Here, H(h,h’) denotes a Zariski open sublocus in fin(CPk,d,h) consisting of maps intersecting transversally with each of the hyperplanes in the basis h, of H 0(09", OCIPM 1)) Since the gluing maps commute with the anti-holomorphic involutions on each of the quotients of the projective varieties WT"(CP'°, d,h), the moduli space of stable maps of genus zero MACH”, d) has a globally well-defined anti—holomorphic involution. We are done. [:1 Corollary 3.1 Let X be a real projective variety having a real structure corresponding to the complex conjugate involution on ClP’k. Then, 37,;(X;t’3) is a real projective variety. proof. It is natural from the construction. See section 5 in [F-P]. D Remark 3.3 1. Corollary 3.1 cannot be extended to any real structures (Cll’l, s), (X, t). Sometimes, the natural set theoretic correspondence f +—-> to f 03 doesn’t define an anti-holomorphic involution on HMX, [3). Let’s consider the case on HACK”, d). An anti-holomorphic involution on 1W"(C1P’k,d) comes from an anti-holomorphic in- volution on the projective variety MACH”, d, h) where the h form a real hyperplane basis. But not every involution on ClP’k allows a real hyperplane basis h. For example, an involution 2 H —1/L~' on Cll”1 does not allow such a basis h. Then, there is no way to make real gluing pieces. 2. The implication of this section is that the moduli space of stable maps of germs 25 zero T7,,(Cll’k,d) is isomorphic to TlTn(CPk,cl)R x3 C, where fin(ClP’k,d)R is a real model. Hence, there is a natural Chow ring homomorphism from Ad(fi,,(ClP’k, d)§‘) to Ad(Hn(ClP’k, d)), induced by complexification of cycles. 3. We introduce an adequate notion of a real group action which gives a natural correspondence in equivariant Chow cycles similar to Remark 3.3 2. We introduce the concept of a real group action on a real scheme X. Definition. Let G be a real Lie group, i.e. Lie group having an antiholomorphic involution, and X be a real scheme. We call a group action G a real group action on X if a morphism u defining a Lie group action u:GXX——>X commutes with real structures, i.e. ,u. is a real morphism. With this notion, we have a natural equivariant Chow ring version’s morphism from A"‘(XR x G2; E G?) to .4*(X x0 BC) by complexification of cycles. We see examples of real group actions. Example 3.2 1. Let T = (C"‘)k+1 act on C1?" in the following way; T x CW —2 CW (th---9tk)'[30;-..;Zk] +——> [tAO-zo;...;t)‘k-zk] Then, it naturally induces a T-action on 71—1,,(le’k, d). We can check it is a real group action. 2. Group actions inducing Cl?" and Oust-(m) are real group actions. i.e., and cxckac 1’2 ck+1xc L}, (t; :0, . . . , 2k; 2) r—i (1‘30, . . . , tzk, t’":) are real group actions. 4 Real part of the moduli space of stable maps and Pro jectivity 4.1 Real part of the moduli space of stable maps We describe the last section’s construction more concretely. All homogeneous co- ordinate forms on a domain curve C1?” in discussion will be standard homogeneous coordinate forms. We will denote any irreducible component as ClP’l for easier look- ing without mentioning a normalization. We may interpret choosing a real ordered hyperplane basis 72. 2 (ho, . . . , 12),) of H0(CPk,OCPk(1)) as choosing a homogeneous coordinate system for ClP’k. Then, the last d(k + 1) marked points in the defini- tion of l—i-rigid stable family gives us some information about the morphism’s nature with polynomials” splitting forms. For example, if we express the hyperplane inter- section points {(Ii,j}0§igk,lgjgd by homogeneous coordinates {[qflj) : QE3)]}OSiSk,ISde, then a degree d morphism f can be expressed with a homogeneous polynomial form [an - Hj:1(q((f}z —q((,3 w) : . . . : ak-nglmgz — (1:11). w)], a,- ¢ 0, where a domain curve is irreducible. we know there is a universal closed locus B in Ivn+d(k+l) through which every morphism from the base scheme 3 of a h-stable family factors. But marked points information doesn’t contain enough data to recover an actual morphism. To recover an exact morphism f, we need to record the ratios a,/ao, i = 1. . . . , It. That could be done by constructing a k-dimensional C*-bundle on a universal closed 10- cus B. Roughly, an associated morphism f with a point ((i'1,...,o:k) in a fibre 27 [H0(ClP’1,7-l1 (8 H51) \ O] x x [H0(CIP’1,7-l;C <8) ”HO—1) \ 0] on a geometric point rep- resenting an irreducible curve (ClP’l. {1)i}1gisn, {[qjy; qf3)]}OS,SkJSJ-Sd) can be thought of as d (1((1) (1 (1(1) (11:1) ),I 1“ [H(‘° - T23) ) 01 “(Z — (2])Ul). . . .,ak H(Z — —(—Zl)ui)] (2) ~2:1 (10,] 1:1 1] ]:1 (1kg , where z — ((1:1)) / qf3))w = w if qf’ljl / (152]) = qflj)/0. Giving an anti-holomorphic involution on the quotient of the moduli space 11—1,,(Cll’k, d, h)/G is just sending (CP1,{a,},:1,,_,,n, [120(z;w);...;pk(z;w)]) to (CPI, {52}2:1,...,n2 [170(.:; w); . . .;1—)k(::;ur)]), where G is a product of symmetric groups (see the proof of Theorem 3.1), 6,- denotes a complex conjugate point of (1,, and p,(z; w) denotes a homogeneous polynomial whose coefficients are the complex conju— gates of those of p,(z; 222). Note that although the polynomial expression depends on a chosen hyperplane basis h, the anti-holomorphic involution defined as above isn’t dependent on the choice of a real ordered basis h because they are related by the PGL(R, k + 1) action which commutes with the anti-holomorphic involution on CP". Thus, we may think of the homogeneous polynomials’ image by an anti-holomorphic involution as their conjugate polynomials regardless of which chosen ordered real hy- perplane basis makes that polynomial expression. The same way of thinking works when we consider reducible curve cases by gluing Operation and restricting our polyno- mial expressions to each irreducible component. Since the quotients of moduli spaces of various h—rigid stable families with real hyperplane basis h cover the moduli space 17,,(C1P’k,d), we may think of a global anti-holomorphic involution on HJCP", d) in the same way. Definition. The '27-th evaluation map €l.’,' is a morphism from .ll,,(ClP’k, d) to ClP’k, sending (Cb,p1, . . . ,p,,,f) to f(p,-). It is easy to see that an evaluation map commutes with an anti-holomorphic involution. The definitions and properties of forgetting maps in Corollary 4.1 can be 28 found in [C-K][7.1.1, 10.1.1]. Corollary 4.1 (i) The evaluation map is a real morphism. (ii) The forgetting morphism from EACH“, (1) to .rTI—n is a real morphism. (iii) The forgetting morphism from EACH“, (1) to TlTn_1(CPk, d) is a real morphism. proof The proof is immediate from the above explanation and the way we gave the anti-holomorphic involution on MACH)", d,h) in Lemma3.5 because that involution commutes with the product of the symmetric group actions and gluing maps, i.e. we can globalize the anti-holomorphic involution to the whole moduli space. [:1 Corollary 4.1 implies that we have corresponding real model maps for the evalu- ation map from HACK”, d)R to RIP" and forgetting maps from MACH”, d)R to ES, from MACE“, d)R to 317n_1(CPk, (1)73. Note that real points MACH”, d)” go to real . —'7'€ pomts Mn . Lemma 4.1 Every point in the real part of A1,,(ClP’k,d), before a compactification, represents a real degree (1 map with real marked points if n 2 3. proof Let (CPl,p1, . . . , p,,, f) be a real point. Then, there is a linear fractional transformation T such that T(p,~) : 1"),- and f o T = f, where f : (CP1,p1,. . .,pn) —> ClP’k is a conjugate map of f. As we have seen in Example 2.1, 3, real points of the Deligne-Mumford moduli space are represented by real pointed curves. By Corollary 4.1 (ii), we see the domain curve (CP1,p1, . . . , p,,) of the map f representing a real point is equivalent to a real pointed curve (ClP’l, r1, . . . , r,,) by a linear fractional trans- formation R such that R(r,~) 2 pi. Note that the conjugation map of a composition map f o R is that map itself because of the number of marked points(n 2 3) in a domain curve. That implies f is a real map. E] Lemma 4.1 can be generalized to any n. Proposition 4.1 Every point in the real part of ilIn(ClP’k, (1) represents a real degree d map with real marked points on the domain curve for any n. 29 proof The domain curve’s marked points condition comes from Corollary 4.1 (iii). The map’s condition follows from Lemma. 4.1 because a forgetting morphism from A—In(CPk, d) to A—1n_1(ClPk, d) is a submersion. [:1 Corollary 4.2 If X is a real projective variety, having a compatible real structure to the complex conjugation map on CW“, then every point in the real part 211,,(X, ,3)” represents a real map with real marked points on the domain curve for any n. proof It is obvious. E] In contrast to the stable map with a nonsingular domain curve case, the image of the stable map with a singular domain curve by a forgetting map to 717,, is not necessarily equivalent to a domain curve of the map because of contractions due to a stability condition. Typical type’s degenerations of domain curves on the real points are: 1. Singular curve with real marked, real gluing points 2. Singular curve with or without components described in 1 and added conjugate pairs of irreducible components without marked points 3. For n = 0 : Singular curve with two irreducible components having a real gluing point such that the gluing point is the unique point in the real part i.e., Singular curve we get by squeezing the equator of the sphere Note that a forgetting map to fin semis all domain curves of type 1 or 2 to the real points of If", by a. contraction. A stable map in the real part 174C111”, 2)” with a singular domain curve having 3 irreducible components CPL, CPL, ClP’g, P.Deligne [Del] constructed is: A component CIR]B has marked points at O, 1, 00 A point 0 in Clip}, is glued to i in CF}, A point 0 in CPA, is glued to —i in Clip]3 An anti-holomorphic involution on a domain curve is given by: 30 ClP’j, ——> CPg, z +—> E ClP’f, —+ OPE,” z »—> E CPL, —> C131,, 2 +—> E A stable map is defined by: Identity maps on CPL, CPI, 1 A zero map on CPR Figure 1: A real h—stable family of maps is a h—stable family of maps which comes from a complexification of an R-scheme stable family of maps, i.e. wcmpj‘i + + pf) 68> [LR*(ORP]¢(3)) is ample on CR, satisfying similar conditions in the h-stable family of maps. The real part of this family of maps consists of stable degree d real maps having marked points on the real parts of the domain curves. Definition. Let CI?" = P0"), where V" = H0(le”k, OCPk(1))- 31 Let h 2 (ho, ..., llk) be an ordered real basis of V“. A real h-stable family of degree (1 maps from n-pointed, genus 0 curves to Cl?" consists of the data: (it 3Cyl: Xe. C —* 3R Xe. Cs {Pi}2=1,...,n,{02,1}2=0,..,k,j=1,...daIt)» where (i) (71?“ : CR —> 83', {pf}, #3) is an R—scheme stable family of degree d maps from n-pointed, R-scheme genus 0 curves to RH)" and 7r comes from a complexification of an R-scheme map n3. (ii) (7r : CR x;.:, C —> 83 X}, C2{1):}2'21.....n2{(1231lizO....k.j:1,...d) is a flat, projective family of n + d(k + 1)-pointed, genus 0, Deligne-Mumford stable curves with sections {[2,} and {(12.2} (iii) For 0 _<_ i S k, there is an equality of Cartier divisors, u*(t.,~) = q“ +q,:,2 + +q,,d Remark 4.1 (a) By base changes, (it : C3 x3,C —> 8K X}; C, {1’1}i=l.....na u) is a stable family of degree (1 maps to CP". (b) Along real points in 83 x3, C, {gig} consists of reals and complex conjugate pairs because each fibre along this locus comes from complexifications of R-scheme maps. More precisely, let f : CIP’l —> ClP’k be a real degree d map. Then, f can be repre- sented by real degree d homogeneous polynomials with standard hyperplane basis of C11“, CPk. Since a chosen basis in the definition of a real h—stable family of maps is real, that basis is related to a standard basis by the PG'LUR, k + 1) action. And we get another real polynomial representation which splits into linear factors. It is obvious that solutions of each homogeneous polynomial consist of reals and complex conjugate pairs. (c) Note that the restrictitm of an anti-holon‘iorphic involution on CR x3, C along real points 8” is complex conjugation maps on each fibre fixing the first 22. marked points. (b) in this remark says the complex line bundles H,- defined by Cartier di- visors u*(h,-), i = 0, . . .,k are real line bundles on each geometric fibre along real points 8” of the base scheme because Cartier divisors ,22.‘(l2,). i = 0, . . . , k are fixed 32 by an anti-holomorphic involution. That means those line bundles come from the complexifications of the line bundles /,1,R*(h§°), i : 0,. . . , k on each fibre. See remark 2.2. ((1) Along the real points 8”, an associated morphism [23' with the last d(k+1) marked points {[qfljl; Qij)]}0gzigk,1gjgd can be related by real homogeneous polynomials d (1) 1) d (1) d [H(z — %<2tr);aj H(z — q—j%u);... elk-(H z — %w)] jzl ‘10,} 3:1 (11,]? 3:1qu , where z— ((1513) /q,if,))w = to if (12(3) Mi? 2 21%) /0. The data (m, . . . ,a'k) can be recorded by constructing a h-dimensional Eff-bundle induced from ;2R*(h§i), i = 0, . . . , k. (e) As we have seen, the topology of 117,,(Cll’k,d) is related to the closed sublocus B in LTMMH). But the topology of real points Mn(Cll’k,d)'e of Mn(CPk,d) doesn’t come from real points of B. It comes from an extended sense’s real locus in B on which we can construct real line bundles to record the additional data ((21, . . . ,ak) in (d). The reason is explained in (b), (c), (d) in this remark. (f) Note that the first n-marked points are on the real points C” of the domain curves along the geometric fibres of real points 8". But the last d(k + 1) marked points are not necessarily on C”. Definition. The derived real h-stable family of degree d maps for a real h-stable family of degree (1 maps from n-pointed, genus 0 curves to ClP’k (71 3 CR Xe. C ‘2 5R Xe; C» {P2}2:1,...,n2 {(12,j}i:0,..,k.j:1,...da #) i5 (7T3 3 CR —> 5”, {P2}2:1,...,m {(Ii.j}i:0,...k,j=1,...da #11:), where CR = ”IT—1(5Tela NR 3 CR —> (319*. Remark 4.2 The construction of real points tends to be geometric because we have to use an extended sense’s real locus B" in B. The following picture helps to under- stand the construction in proposition 4.2. 33 72.691151 H3®H§‘l \ / crk —> ark 7r.(7-l.®7t61) \ tr ivr' / wimfefli”) specC —> speclR Note that r.(’H2 8) H31) '5 721m? 82 725“) as C. Proposition 4.2 There is a universal real sublocus fin(CPk,d,h)" for the derived real h-stable family of degree (1 maps in T7,,(C1Pk,d,l—i). The real part Hn(CPk,d)re of the moduli space of stable maps Wn(CPk,d) is obtained by gluings of the quotient by the product of a symmetric group action G. See section 3.2 [II] for the definition of G. proof Let h be given. The universal closed sublocus B is a real projective variety. The quotient variety B / G has an antiholomorphic involution induced from that of B. Then, the points in the real part (B /G)”6 represent pointed curves with n first real marked points and d(k + 1) last reals and complex conjugate marked points and singular curves described right after Proposition 4.1. BUn+d(k+l) i 7TB B 11) B/G Let’s denote n51 opr‘1((B/G)“’) by BUR,,+d(k+1). When we restrict our attention to the locus BUR-”+41,“ ), ”H, has a natural fibrewise antiholmnorphic automorphism. So does H,- ® 710— 1. This fibrewise antiholomorphic automorphism induces a. fibrewise antiholomorphic automorphism when we consider the direct image sheaf 773..(’H.- <8) 7215‘). See remark 4.2. This allows us to construct real line bundles and then, the desired k-dimensional IR*-l)11ndle W12(CPk,(l,l_2)" over B". 34 To see the constructed space TITn,(CPk, d, h)" is a universal locus for the derived real h—stable family of maps, we observe that the natural morphism from SR XR C to B sends the real points of a real h—stable family to B" due to the types of the last d(k+ 1) marked points. Then, the additional information canonically corresponds to a point in the k-dimensional R*-bundle as described in remark 4.1. This correspondence is consistent with what Fulton-Pandharipande did in [F-P]. We will see a concrete example right after this proof. This locus is preserved by a symmetric group G action. And the moduli space we get by gluing the quotient spaces MR(CPk,d,h)er/G for various h is the real points 717,,(CPk, d)” of the moduli space of stable maps of genus zero. El Example 4.1 We see the concrete case. CP2 i 7T' i W i i 8’6 3 s ——> c E B" Let (CPZ, f) be a real degree 2 map to CPQ, where f(lz; w]) = [new w]); arses; wl);a2f2(lz; w])] = [22 — Mae? +w‘2);a2(222+w?)t where a, E R \ 0. Assume marked points from the hyperplane intersection points {qu} in CPé, i.e. zeros of f,, are given by {{[1; 1], [1; —1]}, {[i; —1], [i: 1]}, {[i; —\/2], [i; \/2]}}. And assume CP}, has marked points {{[i;llali;-1l}a{[-1;-1l,[-1:1]}2{[-1;-\/2—l»[-1;\/§l}}- (i) Then, we see (CPj, {21,-,j}) and (CP;, {qu}) are equivalent by «,9( z;w]) = [iz; w], where go; CP}, —+ CPg. (ii) Note that there is a bundle isomorphism between 71',- ® 712,-] and ’H, ® H51, sending a generator go/g, to a generator 90 o 922/9,- 0 <29 2 fé/fi', where ”HQ, ’H, come from effective Weil divisors (1],, + gag, (1,,1 + gig and [g0([::; w]); g1([z; w]); g2([z; w])] = [—z2 — w2; —z2 + wQ; —2z2 + w2]. That induces an isomorphism between 7T:(Hli ® [Hg—1)\0 = H0((Cll)(l,, H’i (8 715—1) \0 and 7r,(7-t.- (8 710-1) \ 0 = H°(CP,§,’H.- (8) H51) \ 0. Both are isomorphic to C“ and we may consider the numbers a, ( of course, degree 0 polynomial) as elements of 7r:(7't’,- 69 716—1) \0 and 7r,(7-£. <8) 715‘) \ 0, which canonically correspond to each other by an induced isomorphism. The reason is a,- - g, o (,9 = a,- - ,-’, where 020 = 1. It is easy to see that a,- E IR \ 0. The way we extend what we observed with geometric points to the morphism from 8"" to 317,,(CPI‘ , d, h)" is similar to what we saw in section 3.2 [11]. Remark 4.3 1. The general construction for the real part W,,(X, )3)“ comes from the modification of sec.5 in [F-P]. 2. The general procedure to decide the number of connected components of the real points 717,,(CPI‘, d)” is not yet well understood. Note that the number of connected components in the Deligne-Mumford moduli space 317,, is (33—3—12 which is from half of the possible cyclic orderings of marked points, before a compactification. But it is connected after a compactification. 3. Orientability of T7,,(CPk,d)"e is also not yet clear. But we may expect most of the cases are non-orientible even for the degree 0 cases because the Deligne—Mumford moduli space 717”, n 2 5 is non-orientible. Note that Wis is from blowing up four points of CP2 whose corresponding real model is non-orientible. 4.2 Projectivity of the real model MAX, fl)IR Fulton-Pandharipande have shown the moduli space ELACP'CJ) is projective in [F-P]. They used Kollar’s semipositivity approach in [K01] to apply the Nakai- Moishezon criterion to a certain power of a determinant line bundle Det(Q). Ample- ness of Det(Q)” implies the projectivity of 37,,(CPk, d). \Ve summarize the definition 36 of a vector bundle Q on :I\_I,,_((C1P’k,d) in [F -P]. The moduli space HACK” , (1, h) is a fine moduli space equipped with a universal family (ml! —+ WAGE“, (LI-L), {12,-}, [1). Let IF’(L,*) be a projective bundle coming from the projectivization of fibres of L“, where L; = 7r..(w§r(:?:l 1),) <8) p*(0(3l))). We can decide the power 1 which allows a EACH”, (1, h)-canonical embedding e : U —> lP’(L,*) by using Riemann-Roch Theo- rem and Lemma3.3. The morphism p. induces a 37,1(C1P’k , (l, h)-canonical embedding e : U —+ IP’(L;‘) x (CHM and the 72 sections {pi} define it sections {(e 0 pi,” 0 pi)} of IP(L;‘) x ClP’k over W,,(Cll’k,d,h). Let 7r’ denote a natural projection map from IP’(L{) x (Cll’k to Til—”(CPL d, h) and P,- a subscheme defined by the i-th section, Ll’ an embedded U by e. The sum of direct image sheaves 7r:(£m (8) Out) EB 63?:17r1(£’" (8 Opt) of the line bundles .0” along Ll’ and H becomes a vector bundle for sufficiently large m by vanishing of higher direct images. That is the definition of the vector bundle Q. We consider the determinant line bundle det(Q) on Win(CPk, d, h). EACH)", (1) is locally a quotient of TEACH)", (Ll—1.). The induced line bundle Det(Q) is a well-defined line bundle except at the singular points. But we get a well-defined line bundle Det(Q)p by raising the power of p for p large enough. In Lemma 3.5, we showed that the moduli space W,1(C1P’k, d,h) is a real fine moduli space equipped with a real universal family (7r : U —> il—I—H(C1Pk,d,h-), {pi}, [1). As we explained in section 2, that implies that there exists a corresponding real model map (FR : UK —> Til-”(Clipkflflfi, {pf}pF). Fulton-Pandharipande’s construction to show projectivity of 37,,(Cll’kfl) works in this setting and it shows the following Proposition. Proposition 4.3 The real model W,1((Cll"k,d)F of H,,( 4, (resp. H3, H4) becomes a non-orientable( resp. orientable) smooth connected manifold, having a Z/2Z—module version’s fundamental cycle. This big difference in geometric preperties after a com- pactification comes from differences in equivalence relations, i.e. whether it preserves orientations or not. We see more equivalence relation tends to make more conver- gence property and so make the moduli space have a fundamental cycle. An intuitive example for this is when the number of marked points is 4. We may think of the 38 real part of M4 as a circle with 3 points removed, F M4 as two circles with 3 points removed from each. But after the compactification A74 becomes diffeomorphic to a circle and W4 to 6 disjoint closed intervals. We observe that F—M-4 is a generically double cover of 794, but at. the singular divisor, it becomes a 4-uple cover. Generally, the number of inverse images at the compactification divisors are dependent on the number of connected components and the number of marked, gluing points on each component. See [F—Oh, sec.10], [F-Oh-Ohta—Ono] for more detailed descriptions about Fukaya-Oh-Ohta-Ono’s moduli space. 5 The Gromov-Witten invariant and real enumer- ative problems As we have shown in the previous sections, the moduli space of stable maps Tiff—”(A2 [3) is a real moduli space if X is a convex real projective variety, having a real structure corresponding to the complex conjugate involution on CP'". The analysis of the real part of MAX, ,8) and the existence of the fundamental cycle as described in Remark 4.4,1,2 allows us to consider whether there is any way to define a real enumerative invariant on the real part WAX, ,8)“ or on the corresponding real model MAX, mg by using homology and cohomology, or Chow group and Chow ring bilinear pairing. In this section, we assume that the variety X is a homogeneous variety. More detailed properties about the homogeneous variety can be found in [F-P, sec.0.2, sec.7]. Since in most cases, the real part 11—42(43)" is non-orientable, it is natural to consider working with a Z/QZ-module ordinary (co)homology. The invariant on 37,,(X, )3)” can be defined by using the real part maps evf" of evaluation maps, i.e., < [ii—1,.(X, )3)"’], e'z.:’1"‘°-*(£1) U . . . U evf,”(fn.) >, where <, > is the bilinear pairing <, >: H,(.i1,,(X,3)raZ/2Z) x H’mnm’, 3) Z/2Z) ——> Z/‘ZZ, 39 the dimension of H,,(X,/3)r€ is l and the real part map of the evaluation map is evfe : H,,(X,B)"e ——> X”. Note that the Poincare duality doesn’t hold in this case because MAX, [3)” is an orbifold. We can define an invariant on the real model WAX, .3)R by using the real model map eviR of the evaluation map. We use Z-module Chow group and Chow ring’s bilinear pairing < [fin(X, B)“],evR*(£R) . . . ei!§*(£§) >, where <, >2 A[(.’l[n(){', 3)?) X Al(ll[.n(.\r, 5)?) ——) Z, l is the dimension of A7,,(X, 1.3)R and the real model map of the evaluation map is ev?‘ : Hn(.\’,3)R —> XR. This time, the invariant is equal to the usual Gromov- Witten invariant < [Hn(X,g/)],evf(£1)...ev;(£,.) > on M,,(X,l3) coming from the bilinear pairing of the complexifications of cycles in WAX, [3)R. So, the invariant defined in this way cannot have a significance as a real enumerative invariant. To relate the previous sections’ results with the real enumerative problem, we will start with the explanation about why the Gromov-VVitten invariant has an implication in enumerative problems in C-scheme case. Readers can see more explicit details in [F-P, sec.7], [C-K, Chapter 7]. Let E), ..., {n be given classes in a Chow ring A“(X) corresponding to subvarieties F1, ..., R, in general position in X. The Gromov-W’itten invariant 15(51, . . .,€n) = ffinma’i) ev’f(£1) - - ev;(§n) =< [W,,(X,,8)],evf(€1)...ev;({") >, where ev, is an i-th evaluation map, can be well-defined only when ZCodimf‘, is the same as the dimension of the moduli space. Roughly speaking, when it has an enumerative meaning, this invariant counts the number of pointed maps (C421,. . . ,pn; f) such that f..([C]) = [3 and f(p,~) E I}. That is, it counts the number of points in 6131—1011) H . . . fl e'12;1(1“,,). Now, suppose I}, i. : 1,...,n., is a real subscheme in X, i.e. f‘, comes from the complexification of PR in XR‘. Then, eLfl(F,-) is a real subscheme in WAX, 1.3). So, the real subscheme evf1(l‘1) fl . . . fl ev;1(l",,) consists of points preserved by the anti- 40 holomorphic involution on 37,,(X, ,3). Counting real curves for the given enumerative problem will be related to the number of points in evfl (R) H. . .flev;1 (PR) in the real part of 11—1,.(X, 1’3) when cycles meet transversally. But the number of points in the real part is dependent on the choice of the actual cycle representatives because the real number field R is not algebraically closed. Note that those numbers are Z/QZ- module invariant because the complex number field C has a field extension degree 2 over R. Therefore, to relate the previous sections’ results with real enumerative problems, studying the existence of real cycles rationally equivalent to the pull-back of real cycles, meeting transversally at real points 21—1,,(X, )3)“ becomes important. Equality in the Gromov-Witten invariant and the actual numbers of intersection points means curves whose i-th marked points go to the real part of I“, are all real curves, i.e. the given enumerative problem is fully real. 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