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Theoretical Physics IC/93/53 HftTH INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ENUMERATIVE GEOMETRY OF DEL PEZZO SURFACES D. Avritzer INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE IC/93/53 1 Introduction International Atomic Energy Agency and Let H be the Hilbert scheme component parametrizing all specializations of complete intersections of two quadric hypersurfaces in Pn. In jlj it is proved that for n > 2. H is United Nations Educational Scientific and Cultural Organization isomorphic to the grassmannian of pencils of hyperquadrics blown up twice at appropriate smooth subvarieties. The case n = 3 was done in [5j. INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS The aim of this paper is to apply the results of [1] and [5] to enuinerative geometry. The number 52 832 040 of elliptic quartic curves of P3 that meet 16 Sines'in general position; as well as, the number 47 867 287 590 090 of Del Pezzo surfaces in ¥* that meet 26 lines in general position are computed. In particular, the number announced in [5] is ENUMERATIVE GEOMETRY OF DEL PEZZO SURFACES corrected. Let us summarize the contents of the paper. There is a natural rational map, /?, from the grassmannian G of pencils of quadrics to W, assigning to w its base locus /3(ir). The map j3 is not defined along the subvariety B = P" x £7(2, n + 1) of G consisting of pencils with a fixed component. Let Cj,C2 C G be cycles of codimensions ai,« and suppose D. Avritzer * 2 we want to compute the number International Centre for Theoretical Physics, Trieste, Italy. T1 l'"C2 + n2a2 - d = dim(G) ABSTRACT In order that c be enumeratively significant (d. S. Kleiman [3],[4]) we must verify that the 1 Tl; intersection Ci" C2 ' of ri\, (resp. n2) general subvarieties representing Ci (resp. C2) is proper. If, on one hand, the cycle Ci^Cz™3 includes the conditions of going through The number 52 832 040 of eliptic quartic curves of P3 that meet 16 lines in more than 2n — 1 points than Ci^Cj"* fl B = 0 and the intersection is proper so that general position as well as the number 47 867 287 590 090 of Del Pezzo Surfaces in P4 c is the number sought. We compute numbers of this kind in section 3 below. On the that meet 26 lines in general position are computed. To this end an explicit description of other hand, if the cycle Ci is the condition that the base locus intersect a line then every element of B satisfies the condition, i.e., C\ 3 B. It follows that the intersection Ci"1, the Hilbert scheme parametrizing complete intersections of two quadrics in P" in terms of n where n\Q,\ = d is not proper. In particular the immbez- c = JG C\ ' has no enuinerative blowing up of Grassmannians is used. The method applies to the complete intersection of significance. In order to compute the intersection numbers corresponding to cycles of this two quadrics in P", n > 3. kind we have to use the blow up G' of G along B. We do that in sections 4 and 5 blowing up G' along a convenient subvariety if necessary. We establish the notation in the next section and review, for the readers' convenience, some results needed in the sequel and to be published elsewhere. ([I]) MIRAMARE - TRIESTE March 1993 2 Preliminaries Let T be the vector space of linear forms in the variables a.'i,Xj, ...,xn+i, n > 3; let P" denote the dual projective space with tautological sequence, T - 0, where rank C = 1 . Let G[2, n + 1) be the grassmannian of linear spaces of codimension 2 in P™ with tautological sequence Permanent address: Depart amento de Matematica, Universidade Federal de Minas Gerais, 0-fK->.F^g-»0, Caixa Postal 702, 31161 Belo Horizonte, M.G., Brazil. where rank K. = 2. Write G = G(2, S2J7) = G(2, iV) for the grassmannian of pencils of hyperquadrics in P", where 2 N = rank(S-tT) = f " + J = + 3n + 2), Q with tautological sequence 0- 0,. where rank .4 = 2, Let B = Pn x G'(2,n + 1) be the variety consisting of pairs (hyperplane,plane of codimension 2) in P". B imbeds in G as the locus of pencils with a fixed component. Let G' denote the blow-up of G with centre B. Let /?' : G' —• H be the induced C = P(£) = P(M) is the scheme of zeros of C Q and over C we get the following rational ma]). It is proved in [1] that the locus of indeterminacy of /}' is the variety C of diagram of maps of bundles: flags of quadrics of dimension n — 3 contained in n — 2 planes. The main result of [1] is: C - C Theorem 1 Ti is isomorpkic to the blow-up of G' along C. Example 2.1: If n = 4, the generic point of H is the Del Pezzo surface of P\ The subvariety B = P4 x G(2,5) is the locus of indeterminacy of 0 : G = G(2,15) —> H. Let G' be the blow-up of G along B. Let B' = P(jVBG) be the excepcional divisor of G'. The map p : B' —> B has the following geometrical interpretation. Given 1 £ B,i = (Ti,Ta) the fibre of B' over TV consists of all cubic surfaces contained in K\ that contain the line ij n JT2. Consider now the rational map M j3' : G' —• H, induced by 0. Let C C B' be the locus of indeterminacy of /?'. Consider n the restriction of ;> : C —* C, where C C Bis the subvariety of pencils with a fixed component a such that the residual plane falls inside a. Given a point JT£C, the fibre of C' over JT consists of all plane conies in the plane a', the residual intersection corresponding Let V denote coker(C —> Q). With this notation we prove in [1], the following to the pencil 7r. theorem: Let G be the blow-up of G along C and C be the exceptional divisor of G, with Theorem 2 Let Af&G be the normal bundle ofB in G. There is a natural map of bundles, p' : C • —• C. Consider IT' € C and jr = p{ir')- A general point of the fibre of C over r jr' g C is the subscheme defined by a. quartic contained in the 3-space determined by <)>: A BG —> C&K^K® SsM IT, singular along the conic of ir' and moreover having this conic as an as an imbedded component. What Theorem 1 says is that G gives an explicit description of H, i.e., of such that: 4 all specializations of Del Pezzo surfaces of P . In the sequel, we apply these results to 1. offC, § is an isomorphism onto the sub-bundle Einunorativc Geometry. 2 £®\ K® ker(S3M —> S3P), All the geometrical interpretations in the example above come from figuring out ex- 2. on C, 4> drops rank, fitting into the diagram with exact sequence: plicitly what are A'BG, JVC'G'. We state without proofs these resultsjor further reference. 2 The reader is refered to [l] for further details. 0 -> V -+ ABG -. L ® A £ ® Tl ® SiM -+ 0, Let C denote the subvariety of G that consists of the pencils of quadrics with a 2 common component such that the distinguished (n — 2)-plane, axis of the residual pencil where V - \ 1C ® SiQ. of hijH.'rplanes, falls inside the fixed component. On B we have the diagram: 3 Enumerative Calculus on the Grassmannian of Pencils of Quadrics For a fixed point p e P", let p = {A € G such that p 6 base locus of A}. Let us compute the the class of [p] in A2G. Consider the following diagram of exact sequences: over G X I, where Oi denotes the sheaf of regular functions on /. Let s be the map _4 —* Oi{2). If s vanishes over a point in (g,p) € G x /, it means that the fibre of i(A) over g restricted to / vanishes, i.e., I C\ base locus of A ^ 0. (Here i is the map from A to It follows that: 3 [I] = h n c2(A' ® Ci(2)), where h = c,(0(l)). 2 Now, c2(A- ® O,(2)) = 4ft - 2ha(A) + c2(A). Therefore we conclude that: where the fibre of VN-I over a point in G are the quadrics that contain the fixed point Using this, the number of elements of G that meet a lines and go through l> points in P", p e P". We have: with a -\- 2b = d, is given by: [p] = Z(s) = C2(A') n G = c2(A) n G For a fixed line / C P", let L = {A € G such that I C base locus of A}. [L] = Z(s), where .is is the map from A to S V? in tiie following diagram: provided the intersection involved is proper. 2 If a = 0, the degree of the cycie above is clearly 1. Otherwise, the degree of the cycles mentioned above is given by the following lemma. Lemma 1 Let a, b be integers such that a + 26 = d = dim(G),a i- 0. Then: •-2-i N-l-b SaV3 Proof: Let q : P(A) Pw~' and g : P(A) —»Gbe induced by the projection of Let // = IUm(A,S2V2). We have: G x PN^{D P(A)). Consider the following diagram of bundles: Hom(A,S2Vi) ^A'®S2Vi^A'e and therefore, L = Z(s) = c6(H) n G = cfe(X ®A'G> A") = c2(Af.
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