H4Bott_20_01_2009.dvi CUBO A Mathematical Journal Vol.12, No¯ 01, (195–217). March 2010 Projective Squares in P2 and Bott’s Localization Formula† Jacqueline Rojas∗ UFPB-CCEN – Departamento de Matemática, Cidade Universitária, 58051-900, João Pessoa-PB – Brasil email : jacq@mat.ufpb.br Ramón Mendoza UFPE-CCEN – Departamento de Matemática, Cidade Universitária, 50740-540, Recife-PE – Brasil email : ramon@dmat.ufpe.br and Eben da Silva UFRPE/UAST-CCEN – Departamento de Matemática e F́ısica, Fazenda Saco, caixa postal 63, 56900-000, Serra Talhada-PE – Brasil email : eben@uast.ufrpe.br ABSTRACT We give an explicit description of the Hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree 4 in the projective plane that allows us to afford a natural embedding in a product of Grassmann varieties. We also use this description to explain how to apply Bott’s localization formula (introduced in 1967 in Bott’s work [2]) to give an answer for an enumerative question as used by the first time by Ellingsrud and Strømme in [8] to compute the number of twisted cubics on a general Calabi-Yau threefold which is a complete inter- section in some projective space and used later by Kontsevich in [16] to count rational plane curves of degree d passing through 3d − 1 points in general position in the plane. †Dedicated to Israel Vainsencher on the occasion of his 60th birthday. ∗Partially supported by CNPq (Edital Casadinho N o 620108/2008-8) 196 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) RESUMEN En este trabajo, damos una descripción expĺıcita del esquema de Hilbert, que parametriza los subesquemas cerrados de dimensión cero y grado 4 del plano proyectivo, esto nos permite mapear este esquema en un producto de variedades de Grassmann. Usamos dicha construcción, para explicar como se utiliza la fórmula de localización de Bott (introducida en 1967 por Bott en [2]) para responder una pregunta de Geometria Enumerativa, tal como lo hicieron Ellingsrud y Strømme en [8], para calcular cuantas cúbicas torcidas existen en una variedad de Calabi- Yau tri-dimensional, que es una intersección completa en algún espacio proyectivo, y que fue usada posteriormente por Kontsevich en [16], para contar curvas planas racionales de grado d pasando por 3d − 1 puntos en posición general en el plano. Key words and phrases: Hilbert scheme, Bott’s localization formula. Math. Subj. Class.: 14C05, 14N05. 1 Introduction Enumerative geometry has been an active and attractive research subject in math for a long time. A typical problem in enumerative geometry asks for the number of geometric objects of a certain type that satisfy a given set of conditions. For example: 1. Very easy: given two distinct points in the plane, how many lines go through all of them? (the answer - a result from Euclidean Geometry - is clearly one.) 2. Easy: given 2N general lines in the plane, how many N –gons are there with its set of vertices meeting all of them? (easy combinatorial answer: {2N − 1}! = factorial of odd’s numbers between 1 and 2N − 1 (see Section 4).) 3. Medium: how many lines lie on a general cubic surface? (Famous answer: 27.) or how many lines lie on a general quintic threefold? (answer: 2875. Hermann Schubert determined this number explicitly at page 72 in [20], see also the computation at page 281 of Cox-Katz’s book [4].) or in a more general way: how many lines lie on a general hypersurface of degree 2n − 3 in Pn? (answer: see [11]) 4. Hard: given 3d − 1 general points in the plane, how many plane rational curves of degree d pass through all of them? (Answer: N (d). N (d) denotes the Gromov-Witten invariants, they have their origins in physics, in the topological sigma models introduced by Witten in [22]. On the other hand, Kontsevich in [16] found a formula that expresses N (d) in terms of N (e) for e < d, so a single initial datum is required for the recursion, namely, the case d = 1, which correspond to the fact that through two distinct points in the plane pass exactly one line. See Kock-Vainsencher [15] for an elementary introduction and chapter 9 in Cox-Katz’s book [4].) CUBO 12, 1 (2010) Projective Squares in P2 ... 197 In the 19th century, geometers developed a powerful ”calculus” for solving enumerative problems. Their method had no rigorous theoretical foundation, but it worked remarkably well. Justifying their results was the subject of Problem 15th on Hilbert’s famous list. In the 20th century, enumerative geometry has been reconceptualized and made rigorous in terms of intersection theory on parameter spaces (see Fulton [10], Kleiman-Laksov [14] and Kleiman [13] for a survey). So, in order to give a correct answer to an enumerative question, the key issue in the study of parameter spaces is to find a compactification. For example, the Kontsevich’s moduli space of stable maps is used in [16] to calculate N (d). In theorem 1.9 of Nakajima’s book [19] is given an explicit description of the Hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree n in A2. And second, what kind of techniques can be used on a given parameter space to solve an enumerative problem. Usually, the answer to an enumerative problem is reduced to compute Chern classes of some vector bundles. So, for example in [7] Ellingsrud-Göttsche study the Chern and Segre classes of tautological vector bundles on the Hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree n over a smooth and projective surface over the complex numbers. The purpose of this article is to explain how to apply Bott’s localization formula to give an answer to question 2 above when N = 4 using an explicit and elementary description of a parameter space for squares in the plane, that is, the Hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree 4 in P2 2 Notation and Convention For any homogeneous ideal I in the ring C[x0, x1, x2], let Id denote the homogeneous part of degree d , that is, I = ⊕∞ d=0Id. And when we refer to the Hilbert polynomial associated to the closed subscheme determined by the homogeneous ideal I we refer precisely to the Hilbert polynomial associated to the C[x0, x1, x2]-module C[x0, x1, x2]/I (see pg. 51 in Hartshorne’s book [12]). Let I = {f ∈ C[x0, x1, x2] | for each i = 0, 1, 2, there is an Ni such that f · x Ni i ∈ I} be the saturation of the homogeneous ideal I in C[x0, x1, x2]. We say that I is saturated if I = I. Let F denote the vector space of linear forms in the variables x0, x1, x2 and Fd the vector space of homogeneous forms of degree d. Let f1, ..., fs ∈ Fd (f1, ..., fs ∈ C[x0, x1, x2]), we denote by [f1, ..., fs] (〈f1, ..., fs〉) the C-vector space generated by f1, ..., fs in Fd (the ideal generated by f1, ..., fs in C[x0, x1, x2]). For each point p ∈ P2, let F p d denote the linear system of forms of degree d vanishing at the point p. Let Gn(Fd) denote the Grassmann variety parametrizing the n-dimensional vector subspaces of Fd. Set X = G2(F2) be the Grassmannian of pencils of conics in P 2, with tautological sequence 0 −→A−→F2 −→F2 −→ 0 (2.1) 198 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) where A ⊂ F2 denote a subbundle of rank 2 with fiber over π ∈ G2(F2) given by the vector subspace π ⊂F2. 3 An explicit description of Hilb4P2 3.1 Hilbert scheme of points in P2 Let HilbdP2 be the Hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree d in P2. As we have a 1-1 correspondence between saturated homogeneous ideals of C[x0, x1, x2] and closed subschemes of P2 (see Ex. 5.10 of Chapter II in Hartshorne’s book [12]) then we set. HilbdP2 =    I ⊂ C[x0, x1, x2] | I is saturated homogeneous ideal in C[x0, x1, x2] such that the Hilbert polynomial of the C[x0, x1, x2]-module C[x0, x1, x2]/I is equal to d.    . (3.1) It is known that HilbdP2 is nonsingular of dimension 2d (see [9]) and that for each positive integer d it embeds in the Grassmann variety of codimension d subspaces of Fd (see pg. 34 in [1], Lecture 15 in [18]). Having in mind (3.1) we are going to give an explicit description of all saturated homogeneous ideals in Hilb4P2. For those who are interested in the scheme structure in more detail, we recom- mend the reading of [1] and [21]. Naturally as suggested by Bézout’s Theorem we begin with a pair of conics in the plane. 3.2 Quadruplets determined by conics For each π = [q1, q2] ∈ X we can associate the ideal Iπ = 〈q1, q2〉⊂ C[x0, x1, x2] (Iπ is a saturated ideal). The variety determined by Iπ correspond to the intersection of two conics. Thus, we have the following two possibilities:    If gcd(q1, q2) = 1 then according to Bézout’s Theorem the number of intersection points between q1 and q2 should be 2x2=4 points counted with multiplicities. If gcd(q1, q2) 6= 1 then we have that q1 = ℓℓ1, q2 = ℓℓ2 with ℓ, ℓ1 ∈ P ( F ) and ℓ2 ∈ P ( F / [ℓ1] ) . CUBO 12, 1 (2010) Projective Squares in P2 ... 199 So in the general case we have the following pictures: . ........ ......... .......... ............. ................. ....................... ................................................... .................................................................................................................................................................................... ......... ........ ......... .......... ............. ................ . ....................... ......................... .......................... ............................ . ............................ .......................... ......................... ....................... . ................ ............. .......... ......... ........ . ...................... .................... ................... .............. .......................................... ......... . ........ .... ....... ...... . ........ ....... ..... ........ ........ ..... ........ ........ ....... . ......... ........ ..... ......... ......... .. ............ ....... . ............. ............ .......... ........... ......... .......... ............ ............. . .................... ..................... ....................... • • • •q1 q2 gcd(q1, q2) = 1 ℓ BB BB BB BB ℓ1 �� �� �� �� �� �� �� ℓ2 • gcd(q1, q2) 6= 1 Thus it is natural to consider the following subvariety of X. Let Y = { [q1, q2] ∈ X | q1 = ℓℓ1, q2 = ℓℓ2 with ℓ, ℓ1 ∈ P ( F ) and ℓ2 ∈ P ( F / [ℓ1] )} . (3.2) Let p be the intersection point of two lines ℓ1 and ℓ2, then Y can be illustrated as follows: Y =    . .................................................................................................................................................................................. ℓ •p    This figure suggest that we need a cubic form in order to obtain three points on the line ℓ. In the next section we are looking for that cubic form. 3.3 Quadruplets generated in degree three Now, we will describe the cubic homogeneous polynomial f ∈ C[x0, x1, x2], that we need to add to the ideal Iπ = 〈ℓℓ1, ℓℓ2〉 in order to get a quadruplets of points in the plane. 3.1. Lemma. Let I = 〈ℓℓ1, ℓℓ2, f〉 ⊂ C[x0, x1, x2] be an ideal where ℓ, ℓ1 and ℓ2 are linear forms such that [ℓ1, ℓ2] ∈ G2(F) and f 6∈ 〈ℓℓ1, ℓℓ2〉 is a cubic homogeneous polynomial. Then we have that 1. If f 6∈ 〈ℓ〉 and f ∈ 〈ℓ1, ℓ2〉 then I is saturated and the Hilbert polynomial of the variety defined by I is 4. 2. If f 6∈ 〈ℓ〉 and f 6∈ 〈ℓ1, ℓ2〉 then I is saturated and the Hilbert polynomial of the variety defined by I is 3. 3. If f ∈ 〈ℓ〉 then the saturation of I is I = 〈ℓ〉 and the Hilbert polynomial of the variety defined by I is t + 1. Proof. See [21]. 200 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) In the general case we have the following pictures: ℓ • • • BB BB BB BB ℓ1 �� �� �� �� �� �� �� ℓ2 • p q1 = ℓℓ1, q2 = ℓℓ2, f3 ∈ 〈ℓ1, ℓ2〉 \ 〈ℓ〉. ℓ q1 = ℓℓ1, q2 = ℓℓ2, f3 ∈ 〈ℓ〉. We conclude from Lemma 3.1 that a good choice for a cubic form f such that the ideal 〈ℓℓ1, ℓℓ2, f〉 determines a quadruplets of points in the plane, will be to begin with f ∈ 〈ℓ1, ℓ2〉 \ 〈ℓℓ1, ℓℓ2〉. Note that, the vector subspace of cubic forms ℓℓ1·F+ℓℓ2·F is equal to the 5-dimensional vector space ℓ·F p 2 of cubic forms that are multiple of the linear form ℓ times a conic passing through the point p ({p} = ℓ1∩ℓ2). On the other hand, the vector space of cubic forms passing through the point p, F p 3 , have dimension 9, so f ∈ 〈ℓ1, ℓ2〉\〈ℓℓ1, ℓℓ2〉 is varying in a 4-dimensional vector space. Thus we have obtained a P3-bundle E1 over Y (cf. (3.2)). In fact, we can consider E1 embedded in G2(F2) × G6(F3) as follows: E 1 ∋ ([ℓℓ1, ℓℓ2], f ) 7−→ ([ℓℓ1, ℓℓ2], ℓ ·F p 2 + [f ]) ∈ G2(F2) × G6(F3) (3.3) with f ∈ P(F p 3 /(ℓ ·F p 2 )). Note that, to each point ([ℓℓ1, ℓℓ2], f ) ∈ E 1, we can associate the homogeneous ideal 〈ℓℓ1, ℓℓ2, f〉 in C[x0, x1, x2]. Next, we will give a description of those points in E 1 whose associated ideal define a quadruplets in the plane. Certainly, if f ∈ 〈ℓ〉 we do not obtain a quadruplet in the plane (cf. Lemma 3.1). Thus the problem now it is to know when a cubic form f ∈ 〈ℓ1, ℓ2〉\〈ℓℓ1, ℓℓ2〉 will be a multiple of the line ℓ. In fact, we have the following result. 3.2. Lemma. Let W = { [ℓ2, ℓℓ1] ∈ X | [ℓ, ℓ1] ∈ G2(F) } ⊂ Y, which is illustrated as W =    . ............................................................................................................................................................ ℓ • p    where {p} = ℓ ∩ ℓ1. Then we have that 1. If [ℓℓ1, ℓℓ2] ∈ Y \ W then 〈ℓ1, ℓ2〉 ∩ 〈ℓ〉 = 〈ℓℓ1, ℓℓ2〉. Therefore does not exist a cubic form f ∈ 〈ℓ1, ℓ2〉 \ 〈ℓℓ1, ℓℓ2〉 being a multiple of ℓ. 2. If [ℓ2, ℓℓ1] ∈ W then 〈ℓ, ℓ1〉 ∩ 〈ℓ〉 = 〈ℓ〉 and 〈ℓ〉3 = 〈ℓ 2, ℓℓ1〉3 ⊕ [ℓϕ] with ϕ(p) 6= 0, that is, ϕ ∈ F2 \ 〈ℓ, ℓ1〉2. Thus the fiber of E 1 over [ℓ2, ℓℓ1] has exactly one point, does not define a quadruplet and all the others will do. In fact, the locus where f is a multiple of ℓ is given by the following section of E1|W CUBO 12, 1 (2010) Projective Squares in P2 ... 201 E1|W ⊃ W 1 ∋ ([ℓ2, ℓℓ1], ℓϕ) 7−→ ([ℓ 2, ℓℓ1], ℓ ·F2) ∈ G2(F2) × G6(F3) ↑ ↑ W ∋ [ℓ2, ℓℓ1] (3.4) And the saturation of the ideal 〈ℓ2, ℓℓ1, ℓϕ〉 is equal to 〈ℓ〉. 3.4 Quadruplets generated in degree four It Follows from Lemma 3.1 (2.) that it does not help to add any other new generator to the ideal 〈ℓℓ1, ℓℓ2, f〉 in order to get a quadruplets of points in P 2. And from (3.) and Bézout’s Theorem that, it is sufficient to choose a degree four homogeneous polynomial g ∈ C[x0, x1, x2] with g 6∈ 〈ℓ〉. Thus we have obtained a P4-bundle E2 over W1 (cf. (3.4)). In fact, we can consider E2 embedded in G2(F2) × G6(F3) × G11(F4) as follows: E 2 ∋ ([ℓ2, ℓℓ1], ℓϕ, g̃) 7−→ ([ℓ 2, ℓℓ1], ℓ ·F2, ℓ ·F3 + [g]) ∈ G2(F2, ) × G6(F3) × G11(F4) (3.5) with ℓϕ ∈ P(Fp3 /(ℓ ·F p 2 )) and g̃ ∈ P(F4/(ℓ ·F3)). In fact, we have that. 3.3. Lemma. Let I = 〈ℓ, g〉 ⊂ C[x0, x1, x2] be an ideal where ℓ is a linear form and g 6∈ 〈ℓ〉 is a quartic homogeneous polynomial. Then we have that I is saturated and the Hilbert polynomial of the variety defined by I is 4. Proof. See [21]. 4 Enumerative Application Now we are interested in giving an answer to the following enumerative question: How many squares are there with its set of vertices meeting eight general lines? ( • • • • ) More generally, how many N -gons are there with its set of vertices meeting 2N general lines? Note that, each vertex in the N -gon is determined by the intersection of a pair of distinct lines. So, let ℓ1, ..., ℓ2N be 2N general given lines in P 2 and set PN = {N − gons having its vertices in exactly one pair of these distinct lines}. Now, fix 2N + 2 general lines ℓ1, ..., ℓ2N +2 in P 2 and let PN +1,i = {(N + 1) − gons having one vertex over ℓ2N +2 and ℓi} for i = 1, ..., 2N + 1. Note that: • PN +1,i ∩PN +1,j = ∅ for i 6= j; • PN +1,i are in bijection with PN for i = 1, ..., 2N + 1; 202 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) • PN +1 = ⋃2N +1 i=1 PN +1,i. Thus, we have that #(PN +1) = ∑2N +1 i=1 #(PN +1,i) = (2N + 1) · #(PN ). Using induction, we see that #(PN +1) = (2N + 1) ·(2N −1) · ... ·5 · 3 ·1 = {2N + 1}! = the factorial of odd’s numbers between 1 and 2N + 1. Therefore, #(P4) = {7}! = 7 · 5 · 3 · 1 = 105. Next we will use Bott’s localization formula to find the answer to the enumerative problem( • • • • ) on an appropriate parameter space. The Bott’s localization formula that we will apply express the integral of a homogeneous polynomial in the Chern classes of a bundle on a smooth, compact variety with a C∗-action in terms of data given by the induced linear actions on the fiber of the bundle and the tangent bundle in the (isolated) fixed points of the action. In fact, Bott’s residues formula said that. 4.1. Theorem. Let T be a torus and X be a smooth, complete variety with a T -action. Let E1, . . . , Es be T – equivariant vector bundles. Then we have that. ∫ X p(E) ∩ [X] = ∑ F ⊂ XT (πF )∗ ( pT (E|F ) ∩ [F ]T cT dF (NF X) ) (4.1) where • F is a (dimX − dF )-dimensional component of X T ; • X T is the fixed point locus; • p(E) = p(E1, ..., Es) is a homogeneous polynomial of degree dimX in the Chern classes of the bundles E′j s. In fact, p(E) is a weighted homogeneous polynomial in the variables x i j = ci(Ej ), where xij has degree i; • NF X denoted the normal bundle of F in X; • [F ]T is the T -equivariant fundamental class of F ; • cT dF (NF X) denoted the top T -equivariant Chern class of the normal bundle NF X; • pT (E|F ) = p(E1T , ..., EsT ), where EiT denoted the quotient bundles associated to Ei; • (πF )∗ denoted the proper pushforward of the morphism F iF →֒ X πX −→ pt︸ ︷︷ ︸ πF . In spite of the possibly awe-inspiring appearance of (4.1) at first (in part because we do not explain what means each ingredient in the formula), we hope to convince the reader that it is rather simple to apply in practice. See [3], [5], [6] and the elementary exposition in [17] for details. See [8] and chapter 9 in [4] for applications. See also [11] for a computational improvement to Bott’s application that have a close connection with Cauchy’s residue formula. CUBO 12, 1 (2010) Projective Squares in P2 ... 203 4.1 Parameter space for squares Let us consider the following two closed subvarieties of G2(F2)×G6(F3) and G2(F2)×G6(F3)× G11(F4) respectively. For each pencil of conics [q1, q2] ∈ G2(F2) , let C ∈ G6(F3) be the linear system defined as follows C =    q1 ·F + q2 ·F if gcd(q1, q2) = 1, ℓ ·F p 2 + [f ] if q1 = ℓℓ1, q2 = ℓℓ2 with ℓ ∈ P(F), [ℓ1, ℓ2] ∈ G2(F) and f ∈ P(F p 3 /(ℓ ·F p 2 )) where {p} = ℓ1 ∩ ℓ2. (4.2) Let X 1 = { ([q1, q2], C) ∈ G2(F2) × G6(F3) | C is defined as in (4.2) } . (4.3) Now for each ([q1, q2], C) ∈ X 1, let Q ∈ G11(F4) be the linear system defined as follows: Q =    q1 ·F2 + q2 ·F2 if gcd(q1, q2) = 1, ℓ ·F p 3 + f ·F if q1 = ℓℓ1, q2 = ℓℓ2, ℓ ∈ P(F), [ℓ1, ℓ2] ∈ G2(F) and f ∈ P(F p 3 /(ℓ ·F p 2 )) with {p} = ℓ1 ∩ ℓ2 such that f /∈ 〈ℓ〉, ℓ ·F3 + [g] if q1 = ℓℓ1, q2 = ℓℓ2, f = ℓϕ where ϕ ∈F2 \F p 2 and g̃ ∈ P(F4/(ℓ ·F3)). (4.4) Let X 2 = { ([q1, q2], C, Q) ∈ X 1 × G11(F4) | Q is defined as in (4.4) } . (4.5) Follows from (3.3), (4.2) and (4.3) that E1 is a subvariety of X1. In the same way follows from (3.5), (4.4) and (4.5) that E2 is a subvariety of X2. Therefore, we have the following diagram for our parameter space X2. E2 →֒ X2 ւ ց W1 →֒ E1 →֒ X1 ↓ ↓ ↓ W →֒ Y →֒ X (4.6) In fact, it is verified that X1 is the blowup of X along Y with E1 being the exceptional divisor and also that X2 is the blowup of X1 along W1 with E2 being the exceptional divisor (see [1], [21]). On the other hand, for a ∈ C, ([x20, x0(x1 + ax2)], x0 · F2, x0 · F3 + [x 4 1]) are distinct points in X2, but its image in Hilb4P2 is equal to the ideal 〈 x0, x 4 1 〉 . Therefore X2 is not isomorphic to Hilb4P2. Nevertheless, can be verified that X2 is isomorphic to the the blowup of Hilb4P2 along the 6-dimensional subvariety of aligned quadruplets (see [1]) ( 〈 x0, x 4 1 〉 is an aligned quadruplets). 204 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) 5 Divisor of Incidence to a Line Let ℓ be a line in P2 and Dℓ be the hypersurface in X = G2(F2) defined by the condition ℓ ∩ q1 ∩ q2 6= ∅ for [q1, q2] ∈ X. Dℓ =    ℓ • • • •    Let D̃ℓ be the subvariety of ℓ × X defined by D̃ℓ = { (q, π) ∈ ℓ × X | q ∈ base locus of the pencil π } . Note that: • D̃ℓ is a codimension two subvariety of ℓ × X. • The image of D̃ℓ under p2 : ℓ×X −→ X, the projection in the second coordinate, is equal to Dℓ. 5.1 Class of Dℓ Let A be the tautological subbundle of G2(F2) as in (2.1). Let us consider the diagram of natural maps of vector bundles over ℓ × X, A→֒ F2 ց ↓ F2/F • 2 ∼= Oℓ(2) here the fiber F2/F • 2 (q,π) is equal to F2/F2 q. Note that the slant arrow vanishes at (q, π) ∈ ℓ × X if and only if (q, π) ∈ D̃ℓ. Hence we have [D̃ℓ] = (c2(A ∨ ⊗Oℓ(2))) ∩ [ℓ × X] = (c2(A) − 2h · c1(A)) ∩ [ℓ × X], where h = c1(Oℓ(1)). Pushing forward via p2 : ℓ × X −→ X, it follows that [Dℓ] = −2c1(A) ∩ [X]. In fact, p2⋆(c2(p ⋆ 2A) ∩ [ℓ × X]) = c2(A) ∩ p2⋆[ℓ × X] = 0. CUBO 12, 1 (2010) Projective Squares in P2 ... 205 c1(p ⋆ 1Oℓ(1)) ∩ [ℓ × X] = c1(p ⋆ 1Oℓ(1)) ∩ [p ⋆ 1(ℓ)], = p⋆1(c1(Oℓ(1)) ∩ [ℓ]), = p⋆1([pt]), = [pt × X]. then p2⋆(c1(p ⋆ 2A) · c1(p ⋆ 1Oℓ(1)) ∩ [ℓ × X]) = p2⋆(c1(p ⋆ 2A) ∩ [pt × X]), = c1(A) ∩ [X]. A local coordinate check shows that Dℓ contains the blowup center Y (see (3.2)) with multiplicity one. Hence we find the formula for the class of the strict transform in X1, [D (1) ℓ ] = −2c1(A) ∩ [X 1] − [E1]. Similarly, (omitting pullbacks) we get for the succeeding strict transform, [D (2) ℓ ] = −2c1(A) ∩ [X 2] − [E2,1] − [E2]. Here we have omitted the pull-back in A and E2,1 denote the strict transform of E1. Now a solution to the question ( • • • • ) in Section 4 asks us to compute the degree of the self-intersection [D (2) ℓ ]8. Thus from Bott’s formula (cf. (4.1)) we have that. ∫ [X2] [D (2) ℓ ]8 = ∑ F ∫ [F ]T [2cT1 (AF ) + c T 1 (O(E 2,1)F ) + c T 1 (O(E 2)F )] 8 cT dF (NF X2) , (5.1) where dF denotes the codimension of the component F in X. F is a component of X T the locus of fixed points for a suitable torus action, starting at X and following all the way up to X2. 6 Fixed Points at X2 Let V be an n-dimensional complex vector space. Then a general action of C∗ on V is diagonalized, so there is a basis {v1, ..., vn} of V such that t ·vi = λ(t)vi for all t ∈ C. In fact, λ is a character of the group C∗. So λ(t) = twi for some integer wi. We also have an induced action on Gk(V ), the Grassmann variety of k-planes in V, given by t · W = [t · w1, ..., t · wk] for any W = [w1, ..., wk] ∈ Gk(V ). And the fixed points are given by: Wi1,i2,...,ik = [vi1 , vi2 , ..., vik ] where (i1i2...ik) is a k-cicle in Sn, so we have at all ( n k ) fixed points in Gk(V ). 206 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) Consider now the action of C∗ over Fd given by t ◦ x i0 0 x i1 1 x i2 2 = t i0w0+i1w1+i2w2 xi00 x i1 1 x i2 2 with i0 + i1 + i2 = d and extend it by linearity. We also have an induced action on Gn(Fd), X 1 and X2 respectively. According to (4.6) the image of E2 and E1 in X are respectively W and Y. And we also have that X2 \E2 ∼= X1 \W1 and X1 \E1 ∼= X\Y. Let E2,1 ⊂ X2 be the strict transform of E1, then we have that: Fixed points in are in correspondence with fixed points in are in correspondence with fixed points in X2 \ (E2,1 ∪ E2) X1 \ E1 X \ Y Fixed points in are mapped on fixed points in are mapped on fixed points in E2,1 \ E2 E1 \ W1 Y \ W E2 W1 W (6.1) So we will look for fixed points having in mind (6.1). 6.1 Fixed points in X2 \ (E2,1 ∪ E2) If the weights (w0, w1, w2) are sufficiently general, we find the following 6 fixed points in X \ Y: π1 = [x 2 0, x 2 1], π2 = [x 2 0, x1x2], π3 = [x 2 0, x 2 2], π4 = [x0x1, x 2 2], π5 = [x0x2, x 2 1], π6 = [x 2 1, x 2 2]. Since this 6 fixed points lie off Y then they lift (isomorphically) all the way up to X2. So their contribution can be obtained at once, down on X. Of course the exceptional divisors give no contribution here. On the numerator of (5.1) we have for 2cT1 (Aπi ) i = 1, ..., 6, Fixed points in X \ Y Aπi Decomposition of Aπi into eigenspaces 2c T 1 (Aπi ) π1 = [x 2 0, x 2 1] [x 2 0, x 2 1] t 2w0 + t2w1 2(2w0 + 2w1) π2 = [x 2 0, x1x2] [x 2 0, x1x2] t 2w0 + tw1+w2 2(2w0 + w1 + w2) π3 = [x 2 0, x 2 2] [x 2 0, x 2 2] t 2w0 + t2w2 2(2w0 + 2w2) π4 = [x0x1, x 2 2] [x0x1, x 2 2] t w0+w1 + t2w2 2(w0 + w1 + 2w2) π5 = [x0x2, x 2 1] [x0x2, x 2 1] t w0+w2 + t2w1 2(w0 + w2 + 2w1) π6 = [x 2 1, x 2 2] [x 2 1, x 2 2] t 2w1 + t2w2 2(2w1 + 2w2) On the denominator of (5.1) we get Nπi X = Tπi X 2 = Tπi X = F2/Aπi ⊗A ∨ πi . Note that the eigen-decomposition of F2 is given by CUBO 12, 1 (2010) Projective Squares in P2 ... 207 F2 = ∑ 0≤i≤j≤2 twi+wj = t2w0 + tw0+w1 + tw0+w2 + t2w1 + tw1+w2 + t2w2 . Thus for π1 = [x 2 0, x 2 1] we have that Tπ1 X = F2/Aπ1⊗A ∨ π1 = F2/[x 2 0, x 2 1]⊗[x 2 0, x 2 1] ∨ = (tw0+w1 + tw0+w2 + tw1+w2 + t2w2 )(t−2w0 + t−2w1 ). Next we give the eigen-decomposition of Tπi X and c T 8 (Tπi X), for i = 1, ..., 6: π1 = [x 2 0, x 2 1] ↔ (t w0+w1 + tw0+w2 + tw1+w2 + t2w2 )(t−2w0 + t−2w1 ), ↔ (w1 − w0)(w0 − w1)(w2 − w0)(w0 + w2 − 2w1)(w1 + w2 − 2w0)(w2 − w1)(2w2 − 2w0) (2w2 − 2w1), π2 = [x 2 0, x1x2] ↔ (t w0+w1 + tw0+w2 + t2w1 + t2w2 )(t−2w0 + t−(w1+w2)), ↔ (w1 − w0)(w0 − w2)(w2 − w0)(w0 − w1)(2w1 − 2w0)(w1 − w2)(2w2 − 2w0)(w2 − w1), π3 = [x 2 0, x 2 2] ↔ (t w0+w1 + tw0+w2 + t2w1 + tw1+w2 )(t−2w0 + t−2w2 ), ↔ (w2 − w0)(w0 − w2)(w1 − w0)(w0 + w1 − 2w2)(w1 + w2 − 2w0)(w1 − w2)(2w1 − 2w0) (2w1 − 2w2), π4 = [x0x1, x 2 2] ↔ (t 2w0 + tw0+w2 + t2w1 + tw1+w2 )(t−2w2 + t−(w0+w1)), ↔ (w1 − w2)(w2 − w0)(w0 − w2)(w2 − w1)(2w1 − 2w2)(w1 − w0)(2w0 − 2w2)(w0 − w1), π5 = [x0x2, x 2 1] ↔ (t 2w0 + tw0+w1 + tw1+w2 + t2w2 )(t−2w1 + t−(w0+w2)), ↔ (w2 − w1)(w1 − w0)(w0 − w1)(w1 − w2)(2w2 − 2w1)(w2 − w0)(2w0 − 2w1)(w0 − w2), π6 = [x 2 1, x 2 2] ↔ (t 2w0 + tw0+w1 + tw0+w2 + tw1+w2 )(t−2w1 + t−2w2 ), ↔ (w1 − w2)(w2 − w1)(w0 − w2)(w0 + w2 − 2w1)(w1 + w0 − 2w2)(w0 − w1)(2w0 − 2w2) (2w0 − 2w1). So the first six contributions to (5.1) are:    28(2w0+2w1) 8 (w1−w0)(w0−w1)(w2−w0)(w0+w2−2w1)(w1+w2−2w0)(w2−w1)(2w2−2w0)(2w2−2w1) + 28(2w0+w1+w2) 8 (w1−w0)(w0−w2)(w2−w0)(w0−w1)(2w1−2w0)(w1−w2)(2w2−2w0)(w2−w1) + 28(2w0+2w2) 8 (w2−w0)(w0−w2)(w1−w0)(w0+w1−2w2)(w1+w2−2w0)(w1−w2)(2w1−2w0)(2w1−2w2) + 28(w0+w1+2w2) 8 (w1−w2)(w2−w0)(w0−w2)(w2−w1)(2w1−2w2)(w1−w0)(2w0−2w2)(w0−w1) + 28(w0+w2+2w1) 8 (w2−w1)(w1−w0)(w0−w1)(w1−w2)(2w2−2w1)(w2−w0)(2w0−2w1)(w0−w2) + 28(2w1+2w2) 8 (w1−w2)(w2−w1)(w0−w2)(w0+w2−2w1)(w1+w0−2w2)(w0−w1)(2w0−2w2)(2w0−2w1) . 6.2 Fixed points in E2,1 \ E2 Since E2,1 \ E2 is isomorphic to E1 \ W1. Then, we have to look for fixed points on Y \ W (cf. (6.1)). We find after some computation the following 3 fixed points in Y \ W. π7 = [x0x1, x0x2], π8 = [x1x0, x1x2], π9 = [x2x0, x2x1]. (6.2) 208 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) Thus to determine the contributions to (5.1) in this case, we only have to calculate cT8 (Ty1 X 1), cT1 (O(E 1)y1 ) for those fixed points y 1 ∈ E1 lying over πi and 2c T 1 (Aπi ) for i = 7, 8, 9. According to (3.3) the fiber of E1 over [ℓℓ1, ℓℓ2] ∈ Y \ W is given by E 1 [ℓℓ1,ℓℓ2] = P(〈ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2 〉) (6.3) where f indicates classes of f ∈ F p 3 modulo ℓ · F p 2 with {p} = ℓ1 ∩ ℓ2. Note that, ℓ = x0, ℓ1 = x1, ℓ2 = x2 for π7 and so on. And can be verified that ([ℓℓ1, ℓℓ2], ℓ · F p + [f ]) ∈ E1 ⊂ X1 with f ∈ {ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2} are fixed points for the induced action of T = C ∗ on X1. Thus we obtain 3 × 4 = 12 fixed points lying in E1 \ W1. In order to compute the contributions coming from this 12 fixed points to (5.1) we need to determine tangent and normal spaces. Since the exact sequence of C∗–representations 0 → TπY → TπX → (NYX)π → 0 splits, we may write the following decomposition into eigen spaces for [ℓℓ1, ℓℓ2] ∈ Y \ W, (NYX)[ℓℓ1,ℓℓ2] = T[ℓℓ1,ℓℓ2]X −T[ℓℓ1,ℓℓ2]Y = T[ℓℓ1,ℓℓ2]G2(F2) −T[ℓℓ1,ℓℓ2]( ∼=Y︷ ︸︸ ︷ P(F) × G2(F)) = (ℓℓ1 + ℓℓ2) ∨ ⊗ F2−(ℓℓ1+ℓℓ2)︷ ︸︸ ︷ (ℓ2 + ℓ21 + ℓ1ℓ2 + ℓ 2 2)− T[ℓℓ1,ℓℓ2]Y︷ ︸︸ ︷(ℓ1 ℓ + ℓ2 ℓ + ℓ ℓ1 + ℓ ℓ2 ) = ℓ 2 2 ℓℓ1 + ℓ 2 1 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ . (6.4) Note that from (6.3) and (6.4) we have the two descriptions, E 1 [ℓℓ1,ℓℓ2] = P((NYX)[ℓℓ1,ℓℓ2]) = P(〈ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2 〉) and (NYX)[ℓℓ1,ℓℓ2] = ℓ22 ℓℓ1 + ℓ21 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ . We can reconcile this two descriptions noting that to any normal vector ξ as in (6.4) we can associated a curve γt in X with tangent ξ at t = 0 such that γt ∈ X \ Y for t 6= 0, so it lifts to a curve γ1t in X 1 whose tangent at t = 0 give a monomial in {ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2} associated to the normal direction corresponding to ξ as described in the following table: Normal vector ξ Curve with tangent ξ Lifts to a curve in X1 ℓ 2 2 ℓℓ1 = (ℓℓ1) ∨⊗ℓ22 γt = [ℓℓ1 + tℓ 2 2, ℓℓ2] γ 1 t = (γt, ℓℓ2·F + (ℓℓ1 + tℓ 2 2)ℓ + (ℓℓ1 + tℓ 2 2)ℓ1 + tℓ 3 2) ℓ 2 1 ℓℓ2 = (ℓℓ2) ∨⊗ℓ21 γt = [ℓℓ1, ℓℓ2 + tℓ 2 1] γ 1 t = (γt, ℓℓ1·F + (ℓℓ2 + tℓ 2 1)ℓ + (ℓℓ2 + tℓ 2 1)ℓ2 + tℓ 3 1) ℓ1 ℓ = (ℓ)∨⊗ℓ1 γt = [ℓℓ1, ℓℓ2 + tℓ1ℓ2] γ 1 t = (γt, ℓℓ1·F + (ℓℓ2 + tℓ1ℓ2)ℓ + (ℓℓ2 + tℓ1ℓ2)ℓ2 + tℓ 2 1ℓ2) ℓ2 ℓ = (ℓ)∨⊗ℓ2 γt = [ℓℓ1, ℓℓ2 + tℓ 2 2] γ 1 t = (γt, ℓℓ1·F + (ℓℓ2 + tℓ 2 2)ℓ + (ℓℓ2 + tℓ 2 2)ℓ2 + tℓ1ℓ 2 2) (6.5) Determination of cT8 (Ty1 X 1), cT1 (O(E 1)y1 ) for those fixed points y 1 ∈ E1 lying over πi and 2cT1 (Aπi ) for i = 7, 8, 9. CUBO 12, 1 (2010) Projective Squares in P2 ... 209 On the other hand, for any π1 ∈ E1 lying over π ∈ Y, we have that Tπ1 X 1 = Tπ1 E 1 + (NE1 X 1)π1 = Tπ1 E 1 π + TπY + [π 1]. Note that [π1] = OE1 (−1)π1 = O(E 1)π1 . (6.6) Let y = [ℓℓ1, ℓℓ2] ∈ Y\W and y 1 i = (y, fi)∈E 1 with fi ∈{ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2}. So for y 1 1 = (y, ℓ 3 1) ∈ E 1 we have that: Ty11 X 1 = Ty11 E 1 y + TyY + [y 1 1] = Ty11 E 1 y︷ ︸︸ ︷ P([ ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2, ℓ 3 2 ]) +Ty( ∼= Y︷ ︸︸ ︷ P(F) × G∈(F)) + [y 1 1] = ℓ2 ℓ1 + ℓ22 ℓ21 + ℓ32 ℓ31︸ ︷︷ ︸ T y1 1 E1y + ℓ1 ℓ + ℓ2 ℓ + ℓ ℓ1 + ℓ ℓ2︸ ︷︷ ︸ TyY + ℓ21 ℓℓ2︸︷︷︸ O E1 (−1) y1 1 . Ay = ℓℓ1 + ℓℓ2 and O(E 1) (y,ℓ31) = OE1 (−1)(y,ℓ31) = ℓ31 f rom (6.5) ←→ ℓ 2 1 ℓℓ2 . We listed below, the eigen-decomposition of the tangent and first exceptional divisor at each fixed point y1i ∈ E 1, following the description above. Fixed point type for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. Tangent and first exceptional divisor y11 = ([x0x1, x0x2], x 3 1) Ty11 X 1 = t(w2−w1) + t(2w2−2w1) + t(3w2−3w1) + t(w1−w0)+ t(w2−w0) + t(w0−w1) + t(w0−w2) + t(2w1−w0−w2), O(E1)y11 = t(2w1−w0−w2). y12 = ([x0x1, x0x2], x 2 1x2) Ty12 X 1 = t(w1−w2) + t(w2−w1) + t(2w2−2w1) + t(w1−w0)+ t(w2−w0) + t(w0−w1) + t(w0−w2) + t(w1−w0), O(E1)y12 = t(w1−w0). y13 = ([x0x1, x0x2], x1x 2 2) Ty13 X 1 = permute w1 and w2 in Ty12 X 1, O(E1)y13 = t(w2−w0). y14 = ([x0x1, x0x2], x 3 2) Ty14 X 1 = permute w1 and w2 in Ty11 X 1, O(E1)y14 = t(2w2−w0−w1). Thus the contribution to (5.1) at each y1i ∈ E 1 lying over y = [ℓℓ1, ℓℓ2] ∈ Y \W is given by: 210 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) Fixed point for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. Contribution to the numerator in (5.1) Contribution to the denominator in (5.1) y11 = (y, x 3 1 ) 2cT1 (Ay) = 2(2w0 + w1 + w2)+ c T 1 (O(E 1) y1 1 ) = 2w1 − w0 − w2. c T 8 (Ty1 1 X 1) = (w2 − w1)(2w2 − 2w1)(3w2 − 3w1)· (w1 − w0)(w2 − w0)(w0 − w1)· (w0 − w2)(2w1 − w0 − w2). y12 = (y, x 2 1x2) 2cT1 (Ay) = 2(2w0 + w1 + w2)+ c T 1 (O(E 1) y1 2 ) = w1 − w0. c T 8 (Ty1 2 X 1) = (w1 − w2)w2 − w1)(2w2 − 2w1)(w1 − w0)· (w2 − w0)(w0 − w1)(w0 − w2)(w1 − w0). y13 = (y, x1x 2 2) permute w1 and w2 in 2cT2 (Ay) + c T 1 (O(E 1) y1 2 ). c T 8 (Ty1 3 X 1) = permute w1 and w2 in c T 8 (Ty1 2 X 1). y14 = (y, x 3 2) permute w1 and w2 in 2cT2 (Ay) + c T 1 (O(E 1) y1 1 ). c T 8 (Ty1 4 X 1) = permute w1 and w2 in c T 8 (Ty1 1 X 1). In fact, if we make a cyclic permutation of x′is in the table above, we will obtain all the 12 contributions to (5.1) determined by the 3 fixed points πi ∈ Y \W for i = 7, 8, 9 (cf. (6.2)). 6.3 Fixed points in E2 According to (6.1) we have to look for fixed points in W. After some computation we find the following 6 fixed points in W: π10 = [x 2 0, x0x1], π11 = [x 2 0, x0x2], π12 = [x 2 1, x1x0], π13 = [x 2 1, x1x2], π14 = [x 2 2, x2x0], π15 = [x 2 2, x2x1]. (6.7) In order to determine the contributions to (5.1) in this case we have to calculate cT8 (Tπ2 X 2), cT1 (O(E 2)π2 ) for those fixed points π 2 ∈ E2 lying over fixed points π1 ∈ W1 ⊂ E1, cT1 (O(E 1)π1 ) for those fixed points π1 ∈ E1 lying over πi and 2c T 1 (Aπi ) for i = 10, ..., 15. We have from (3.3) that the fiber of E1 over [ℓ2, ℓℓ1] ∈ W is given by: E 1 [ℓ2,ℓℓ1] = P(〈ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 〉) (6.8) where f indicates classes of f ∈F p 3 modulo ℓ ·F p 2 with {p} = ℓ∩ℓ1. Note that, ℓ = x0, ℓ1 = x1 for π10 and so on. And can be verified that ([ℓ 2, ℓℓ1], ℓ·F p +[f ]) ∈ E1 ⊂ X1 with f ∈{ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2} are fixed points for the induced action of T = C∗ on X1. Thus we obtain: { 6 × 3 = 18 fixed points lying in E1 \ W1 if f ∈{ℓ31, ℓ 2 1ℓ2, ℓ1ℓ 2 2}, 6 × 1 = 6 fixed points lying in W1 if f = ℓℓ22. (6.9) In order to compute the contribution of this fixed points to (5.1) we need to determine tangent and normal spaces as we did in (6.4). CUBO 12, 1 (2010) Projective Squares in P2 ... 211 We may write the following decomposition into eigen spaces for [ℓ2, ℓℓ1] ∈ W, (NYX)[ℓ2,ℓℓ1] = T[ℓ2,ℓℓ1]X −T[ℓ2,ℓℓ1]Y = T[ℓ2,ℓℓ1]G2(F2) −T[ℓ2,ℓℓ1]( ∼=Y︷ ︸︸ ︷ P(F) × G2(F)) = (ℓ2 + ℓℓ1) ∨ ⊗ F2−(ℓ 2+ℓℓ1)︷ ︸︸ ︷ (ℓℓ2 + ℓ 2 1 + ℓ1ℓ2 + ℓ 2 2)− T [ℓ2,ℓℓ1] Y ︷ ︸︸ ︷(ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1 ) = ℓ 2 1 ℓ2 + ℓ1ℓ2 ℓ2 + ℓ 2 2 ℓ2 + ℓ 2 2 ℓℓ1 . (6.10) Note that from (6.8) and (6.10) we have the two descriptions, E 1 [ℓ2,ℓℓ1] = P((NYX)[ℓ2,ℓℓ1]) = P(〈ℓℓ 2 2, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 〉) and (NYX)[ℓ2,ℓℓ1] = ℓ21 ℓ2 + ℓ1ℓ2 ℓ2 + ℓ22 ℓ2 + ℓ22 ℓℓ1 . Again we can reconcile this two descriptions as we did in (6.5). In this case the correspondence is given by: ℓ 2 1 ℓ2 ℓ1ℓ2 ℓ2 ℓ 2 2 ℓ2 ℓ 2 2 ℓℓ1 l l l l ℓ31 ℓ 2 1ℓ2 ℓ1ℓ 2 2 ℓℓ 2 2 (6.11) Now, let w1i = ([ℓ 2, ℓℓ1], fi) ∈ E 1 with fi ∈{ℓℓ 2 2, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2}. Contributions to (5.1) coming from w1i for i = 2, 3, 4 Note that the three points w1i for i = 2, 3, 4 lift (isomorphically) all the way up to X 2 since X2 \ E2 ∼= X1 \ W1. So their contribution can be obtained at once on X1. So for w12 = (w, ℓ 3 1) ∈ E 1 lying over w = [ℓ2, ℓℓ1] we have from (6.6) that Tw12 X 1 = Tw12 E 1 w + TwY + [w 1 2] = Tw12 E 1 w︷ ︸︸ ︷ P([ ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 ]) +Tw( ∼= Y︷ ︸︸ ︷ P(F) × G∈(F)) + [w 1 2] = ℓℓ22 ℓ31 + ℓ2 ℓ1 + ℓ22 ℓ21︸ ︷︷ ︸ T w1 2 E1w + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸ ︷︷ ︸ TwY + ℓ21 ℓ2︸︷︷︸ O E1 (−1) w1 2 . (6.12) A[ℓ2,ℓℓ1] = ℓ 2 + ℓℓ1 and O(E 1) ([ℓ2,ℓℓ1],ℓ 3 1) = OE1 (−1)([ℓ2,ℓℓ1],ℓ31) = ℓ31 f rom (6.11) ←→ ℓ 2 1 ℓ2 . We listed below, the eigen-decomposition of the tangent and first exceptional divisor at each fixed point w1i ∈ E 1 for i = 2, 3, 4, following the description above. 212 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) Fixed point for ℓ = x0 and ℓ1 = x1. Tangent and first exceptional divisor w12 = ([x 2 0, x0x1], x 3 1) Tw12 X 1 = t(w0+2w2−3w1) + t(w2−w1) + t(2w2−2w1) + t(w1−w0)+ t(w2−w0) + t(w2−w0) + t(w2−w1) + t(2w1−2w0), O(E1)w12 = t(2w1−2w0). w13 = ([x 2 0, x0x1], x 2 1x2) Tw13 X 1 = t(w0+w2−2w1) + t(w1−w2) + t(w2−w1) + t(w1−w0)+ t(w2−w0) + t(w2−w0) + t(w2−w1) + t(w1+w2−2w0), O(E1)w13 = t(w1+w2−2w0). w14 = ([x 2 0, x0x1], x1x 2 2) Tw14 X 1 = t(w0−w1) + t(2w1−2w2) + t(w1−w2) + t(w1−w0)+ t(w2−w0) + t(w2−w0) + t(w2−w1) + t(2w2−2w0), O(E1)w14 = t(2w2−2w0). Thus the contribution to (5.1) at each w1i ∈ E 1 for i = 2, 3, 4 lying over w = [ℓ2, ℓℓ1] is given by: Fixed point for ℓ = x0, ℓ1 = x1. Contribution to the numerator in (5.1) 2c T 1 (Aw)︸ ︷︷ ︸ 2(3w0+w1) + cT1 (O(E 1) w1 i ) Contribution to the denominator in (5.1) c T 8 (Tw1 i X 1) w12 = (w, x 3 1) 2(3w0 + w1) + (2w1 − 2w0) 4(w0 + 2w2 − 3w1)(w2 − w1) 3(w1 − w0) 2(w2 − w0) 2 w13 = (w, x 2 1x2) 2(3w0 + w1) + (w1 + w2 − 2w0) −(w0 + w2 − 2w1)(w2 − w1) 3(w1 − w0)(w2 − w0) 2 (w1 + w2 − 2w0) w14 = (w, x1x 2 2) 2(3w0 + w1) + (2w2 − 2w0) 4(w0 − w1) 2(w1 − w2) 3(w2 − w0) 3 Making a cyclic permutation of x′is in the table above, we will obtain all the 18 contribu- tions to (5.1) determined by the 6 fixed points πi ∈ W for i = 10, ..., 15 (cf. (6.7) and (6.9)). Contributions to (5.1) coming from w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2) ∈ W 1 In order to determine the contributions to (5.1) in this case we have to calculate cT8 (Tw2 X 2), cT1 (O(E 2)w2 ) for those fixed points w 2 ∈ E2 lying over the fixed point w11 ∈ W 1 ⊂ E1, cT1 (O(E 1)w11 ) and 2cT1 (Aw). Consider now the fiber of E2 over w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2). According to (3.5), it is just E 2 ([ℓ2,ℓℓ1],ℓ̃ℓ 2 2) = P(〈 ℓ̃41, ℓ̃ 3 1ℓ2, ℓ̃ 2 1ℓ 2 2, ℓ̃1ℓ 3 2, ℓ̃ 4 2 〉) (6.13) where g̃ indicates classes of g ∈F4 modulo ℓ·F3. And can be verified that ([ℓ 2, ℓℓ1], ℓ·F2, ℓ·F2+[g]) ∈ E2 ⊂ X2 with g ∈{ℓ41, ℓ 3 1ℓ2, ℓ 2 1ℓ 2 2, ℓ1ℓ 3 2, ℓ 4 2} are fixed points for the induced action of T = C ∗ on X2. Thus we obtain 6 × 5 = 30 fixed points lying in E2. In order to compute the contribution of this 30 fixed points to (5.1) we need to determine tangent and normal spaces as we did in (6.4). CUBO 12, 1 (2010) Projective Squares in P2 ... 213 We may write the following decomposition into eigen spaces for w11 = (w, ℓℓ 2 2) ∈ W 1 lying over w = [ℓ2, ℓℓ1], (NW1 X 1)w11 = Tw11 X 1 −Tw11 W 1 = Tw11 E 1 w + TwY + [w 1 1] −Tw11 W 1 = Tw11 E 1 w︷ ︸︸ ︷ P([ ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 ]) +Tw( ∼= Y︷ ︸︸ ︷ P(F) × G∈(F)) + [w 1 1] −Tw11 W 1 = ℓ31 ℓℓ22 + ℓ21 ℓℓ2 + ℓ1 ℓ ︸ ︷︷ ︸ T w1 1 E1w + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸ ︷︷ ︸ TwY + ℓ22 ℓℓ1︸︷︷︸ O E1 (−1) w1 1 − ( ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ1 ) ︸ ︷︷ ︸ T w1 1 W1 = ℓ 3 1 ℓℓ22 + ℓ 2 1 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ + ℓ 2 2 ℓℓ1 . (6.14) Note that from (6.13) and (6.14) we have the two descriptions at w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2), E 2 w11 = P((NW1 X 1)w11 ) = P(〈 ℓ̃ 4 1, ℓ̃ 3 1ℓ2, ℓ̃ 2 1ℓ 2 2, ℓ̃1ℓ 3 2, ℓ̃ 4 2 〉) and (NW1 X 1)w11 = ℓ31 ℓℓ22 + ℓ21 ℓℓ2 + ℓ1 ℓ + ℓ2 ℓ + ℓ22 ℓℓ1 . We can reconcile this two descriptions as we did in (6.5) and (6.11). In this case the correspondence is given by: ℓ 3 1 ℓℓ22 ℓ 2 1 ℓℓ2 ℓ1 ℓ ℓ2 ℓ ℓ 2 2 ℓℓ1 l l l l l ℓ41 ℓ 3 1ℓ2 ℓ 2 1ℓ 2 2 ℓ1ℓ 3 2 ℓ 4 2 (6.15) Now let w2i = (w 1 1, g̃i) ∈ E 2 with gi ∈ {ℓ 4 1, ℓ 3 1ℓ2, ℓ 2 1ℓ 2 2, ℓ1ℓ 3 2, ℓ 4 2}. For w 2 1 = (w 1 1, ℓ̃ 4 1) ∈ E 2 lying over w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2) we have from (6.6) changing 1 by 2 that: Tw21 X 2 = Tw21 E 2 w1 1 + Tw11 W 1 + [w21] = ℓ31ℓ2 ℓ41 + ℓ22 ℓ21 + ℓ32 ℓ31 + ℓ42 ℓ41︸ ︷︷ ︸ T w2 1 E2 w1 1 + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸ ︷︷ ︸ T w1 1 W1 + ℓ31 ℓℓ22︸︷︷︸ O E2 (−1)w2 1 . O(E2)w21 = OE2 (−1)(w11,ℓ̃41) = ℓ41 ( f rom (6.15) ←→ ℓ 3 1 ℓℓ22 ). We listed below, the eigen-decomposition of the tangent and first exceptional divisor at each fixed point w2i ∈ E 2 for i = 1, ..., 5, following the description above. 214 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) Fixed point for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. Tangent and second exceptional divisor w21 = (w 1 1, x̃ 4 1) Tw21 X 2 = t(w2−w1) + t(2w2−2w1) + t(3w2−3w1) + t(4w2−4w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(3w1−w0−2w2), O(E2)w21 = t(3w1−w0−2w2). w22 = (w 1 1, x̃ 3 1x2) Tw22 X 2 = t(w1−w2) + t(w2−w1) + t(2w2−2w1) + t(3w2−3w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(2w1−w0−w2), O(E2)w22 = t(2w1−w0−w2). w23 = (w 1 1, x̃ 2 1x 2 2) Tw23 X 2 = t(2w1−2w2) + t(w1−w2) + t(w2−w1) + t(2w2−2w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(w1−w0), O(E2)w23 = t(w1−w0). w24 = (w 1 1, x̃1x 3 2) Tw24 X 2 = t(3w1−3w2) + t(2w1−2w2) + t(w1−w2) + t(w2−w1)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(w2−w0), O(E2)w24 = t(w2−w0). w25 = (w 1 1, x̃ 4 2) Tw25 X 2 = t(4w1−4w2) + t(3w1−3w2) + t(2w1−2w2) + t(w1−w2)+ t(w1−w0) + t(w2−w0) + t(w2−w1) + t(2w2−w0−w1), O(E2)w25 = t(2w2−w0−w1). Following the description given in (6.12), we obtain the following eigen-decomposition for the tangent space of X1 at w11. Tw11 X 1 = Tw11 E 1 w + TwY + [w 1 1] = Tw11 E 1 w︷ ︸︸ ︷ P([ ℓℓ22, ℓ 3 1, ℓ 2 1ℓ2, ℓ1ℓ 2 2 ]) +Tw( ∼= Y︷ ︸︸ ︷ P(F) × G∈(F)) + [w 1 1] = ℓ31 ℓℓ22 + ℓ21 ℓℓ2 + ℓ1 ℓ ︸ ︷︷ ︸ T w1 1 E1w + ℓ1 ℓ + ℓ2 ℓ + ℓ2 ℓ + ℓ2 ℓ1︸ ︷︷ ︸ TwY + ℓ22 ℓℓ1︸︷︷︸ O E1 (−1) w1 1 . (6.16) and O(E1) ([ℓ2,ℓℓ1],ℓℓ 2 2) = OE1 (−1)([ℓ2,ℓℓ1],ℓℓ22) = ℓℓ22 f rom (6.11) ←→ ℓ 2 2 ℓℓ1 , then cT1 (O(E 1)w11 ) = 2w2−w0−w1 doing ℓ = x0, ℓ1 = x1 and ℓ2 = x2. Thus the contribution to (5.1) at each w2i ∈ E 2 for i = 1, ..., 5 lying over w11 is given by: CUBO 12, 1 (2010) Projective Squares in P2 ... 215 Fixed point for ℓ = x0, ℓ1 = x1 and ℓ2 = x2. Contribution to the numerator in (5.1) 2c T 1 (Aw)︸ ︷︷ ︸ 2(3w0+w1) + c T 1 (O(E 1 ) w1 1 ) ︸ ︷︷ ︸ 2w2−w0−w1 + cT1 (O(E 2) w2 i ) Contribution to the denominator in (5.1) c T 8 (Tw2 i X 2) w21 = (w 1 1, x̃ 4 1) (5w0 + w1 + 2w2) + (3w1 − w0 − 2w2) 24(w2 − w1) 5(w1 − w0)(w2 − w0)(3w1 − w0 − 2w2) w22 = (w 1 1, x̃ 3 1x2) (5w0 + w1 + 2w2) + (2w1 − w0 − w2) −6(w2 − w1) 5(w1 − w0)(w2 − w0)(2w1 − w0 − w2) w23 = (w 1 1, x̃ 2 1x 2 2) (5w0 + w1 + 2w2) + (w1 − w0) 4(w2 − w1) 5(w1 − w0) 2(w2 − w0) w24 = (w 1 1, x̃1x 3 2) (5w0 + w1 + 2w2) + (w2 − w0) −6(w2 − w1) 5(w1 − w0)(w2 − w0) 2 w25 = (w 1 1, x̃ 4 2) (5w0 + w1 + 2w2) + (2w2 − w0 − w1) 24(w2 − w1) 5(w1 − w0)(w2 − w0)(2w2 − w0 − w1) Making a cyclic permutation of x′is in the table above, we will obtain all the 30 contributions to (5.1) determined by the 6 fixed points w11 = ([ℓ 2, ℓℓ1], ℓℓ 2 2) ∈ W 1 (cf. (6.7) and (6.9)). So, there are altogether 66 fixed points as indicated bellow by the bold points. In fact, consider the diagram below, where we use ”•” to indicate the terminal fixed points and ”◦” to indicate the non-terminal ones. E2 ∋    • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ↓ W1 ∋ { ◦ ◦ ◦ ◦ ◦ ◦ E1 \ W1 ∋    • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •    ∈ E1 \ W1 ↓ ↓ ◦ ◦ ◦ ◦ ◦ ◦ ︸ ︷︷ ︸ W ◦ ◦ ◦ ︸ ︷︷ ︸ Y\W • • • • • • ︸ ︷︷ ︸ X\Y In the first line in the bottom we put the 15 = 6 (in X \ Y)︸ ︷︷ ︸ terminal + 3 (in Y \ W)︸ ︷︷ ︸ non-terminal + 6 (in W)︸ ︷︷ ︸ non-terminal fixed points in X. In the middle, we have 12 (respectively 18) terminal fixed points in E1 \ W1 that are mapped to the 3 (respectively 6) fixed points in Y \ W (respectively W) by the the first blow-up map, and we also have 6 non-terminal fixed points in W1 that are mapped to the 6 fixed points in W (W1 is the second blow-up center and W1 ∼= W). 216 Jacqueline Rojas, Ramón Mendoza and Eben da Silva CUBO 12, 1 (2010) At the top, we have 30 terminal fixed points in E2 that are mapped to the 6 fixed points in W1 by the the second blow-up map (this last 6 fixed points in W1 are mapped isomorphically to the 6 fixed points in W by the first blow-up map). Finally using a MAPLE script, we find one more time that there exist 105 squares whose vertices lie over 8 lines in general position in P2. Acknowledgements The first and second author wish to express their gratitude to Instituto Nacional de Matemática Pura e Aplicada (IMPA) and to Departamento de Matemática da Universidade Federal da Paráıba - Campus 1, for providing the right environment for concluding this work. The first author was partially supported by Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico (CNPq). Received: July, 2008. Revised: January, 2009. References [1] Avritzer, D. and Vainsencher, I., Hilb4P2, in Proceedings of the Conference at Sitges, Spain (1987), ed. S. Xambó, Springer-Verlag Lect. Notes Math. 1436 (1987), 30–59. [2] Bott, R., A residue formula for holomorphic vector fields, J. Differential Geom. 1 (1967), 311–330. [3] Brion, M., Equivariant cohomology and equivariant intersection theory, in Representation theory and algebraic geometry, Kluwer (1998), 1–37. [4] Cox, D. and Katz, S., Mirror symmetry and algebraic geometry, Math. Surv. 68 - Amer. Math. Soc., Providence, RI, 1999. 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