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. The number of elements of G that contain a lines and go through b points in P", with 6a 4- 1h —

/ c2{A?«ci{A? = I c7{Af\ JQ, JG provided the intersection involved is proper, hy a counting of constants argument as in S. We get the diagram below: Kidman, [2]. We give some specific examples below. Before we clo that let us calculate one more geometric cycle. For a fixed line7 C P", O (~\) = q'( let. 1 = {A e G such that / n base locus of A ^ 0}._Let I = {(A,p), such that p € A base locus of A} C GxIcG xP°. One checks that I maps birationally to 1. Consider the diagram: g-(A) g'(B)

q'(T) —> g*( 0t{2) st where the second vertical exact sequence is the pull-back by q of the tautological sequence 4 Enumerative Calculus on the I Blow-up N of P -\ It follows that CU(-l) = q-(Oe(~l)). m Set V = OA(-\) and t, = Ci(V'); U = Or(-1) and u = cY{U ). We have: We keep the notation of previous sections. Let G' be the blow-up of G along B. We have the diagram: C(.A) = 1 - u - u + uu and S(T) = (S(V))"1 = 1 - v, where C'(.4)(resp.,'i(T)) denotes the total Chern(resp. Segre) polynomial of the bundles B' ^ G' involved. We get: l (-2c,(A))aC2(A)b = / ^ (" + v)°(uv)brv. G p B ^ G, Since P(.4) = P(

/ (u + v)°(uv)b2*v = [ 2a H,,p+W~i where B' is the exceptional divisor in G'. We consider, as before, 1 C G the cycle of P(T) JP(T) pencils of quadrics whose base locus intersect a fixed line I. Then we have: Recall P(T) is a PA'"2 bundle over P"'1. Therefore S(P(7)) = Ei>n-2 ?.(«')• As a N 2 v 1 V+! Lemma 2 Let V be the strict transform of 1 in G'. The cycle 1 contains B with multi- consequence we get q.u ' = l,f/,ic' ~ = —i;,q,H = 0,! > 0. The degree of the cycles plicity one and, therefore, we have: above can therefore be computed as:

2' N -2-6 A' - 1 - 6 Now a + 2/> = d - dim{G) = 2N - i. So, a + 2b - N + Z = N - I and the lemma is Proof: Let U = A2' be a standard open neighborhood of IT = (Aj, XiAj) g G (where proved. 2 t = l/2(« + 3n -2)) with coordinate functions a, (i = 1,2, ...,t) and b} (j - 1,2,....<) so Example 3. l:]f u = 3, [l]"[p]\ the number of elliptic quartic curves that that meet a lines that the restriction of the tautological bundle A to U is trivial with basis: ai\d go through b points, with n + 2h = d — 16 can be calculated, using the method above, 2 provided /) > G. Within theso bounds [l]'[p]'nB = 0, and the intersection is proper. Since ... + ajn_,A2Xn+! + ... + atXn+x •V =10, we get: v(2) = A-iA-j + MV + 62A',A'3 + ... + ^-IAJAV A general point in U may be thought of as the 2x1 matrix:

1 0 a2 4 c Example 3.2:lf ?? = 4. [l]°[p] [L) , the number of Del Pezzo Surfaces that that meet a 0 1 b2 lines, go through b points and contain c lines, with a + 2b + 6c = d = 26, can be calculated, using the method above, provided b + 3c > 8. Within these bouuds [l]"[p]l'[L]<: n B = 0, Let 1 C P" be the line given by A'3 = Xt = ... = An+] = 0. The condition that 1 and the intersection is proper. Since iV = 15, we get: intersects an element of U is given by:

1 0 aiX{ x, b\ A',2 0 0 A', 6i A',2

T " ' 13 - (6 + 3c) ) { 14-(6 +3c) Since all the Xt cannot vanish at the same time it follows that A'i / 0. If, liowcvei, in the examples above we were interested in smaller values of b then we Now let Go be an affine subspace of G n U defined by bt = b2 = • -. = &2n-i = 0. could not use the formula of Lemma 1, for in this case the intersection would not be We have that B n Go is given by (alla2,... ,at), as schemes.{See [1],[5].) It follows that 1 3 B with multiplicity one. proper. We need to blow up G along B. We compute numbers of this kind in the next Let p' = {A' £ G' such that p 6 base locus of A'}. Calculating the expression of [p'] section. in A2G' we find that [p'j = IT*[p]. The number of complete intersections of two quadrics

8 that meet « lines and go through h points, with a -f 2b = d, can therefore be calculated Lemma 3 The class o in the Grofhendick ring o/B is given by: as: b (-2cl{A)-B)-c-i(A) , JG' provided the intersection is proper. Let us verify when the intersection above is proper. We recall that p : B' —> B has the following geometrical interpretation. Given IT £ B = P" x G{2,n + Proof: A'BG fits into the exact sequence 0 -» TB -> TG |B -+ A'BG -• 0. In the Grothendick Ring of B we can write TB = 1), w = ( IT, . ir2), the fibre of B' over IT consists of all cubic hypersurfaces contained in V\ / t t ® M + K ® Q. We also have: that contain jrt D ir2- It follows that if b > n + 1 the intersect ion [l ]°[p'] ' is proper for it avoids B'. TG |B= (-4 © B) |B= t ® /C ® (S2T -L%K)- rank{S2T)l O AJ. Let us compute for b > n + 1 the intersection Therefore we get: = N(t ©/£)-(„ + l)£ - (n + 1)£. Before we do specific examples we note that since B = P"x (7(2, u + 1). we have: In ordfi to compute the degree of the zero cycle above we need to compute: i'cj(A) = Cj(i'(A)} = Cy(C ® AC), keeping the notation of section 1. / ( Example 4.2: Let us compute the number of Del Pezzo Surfaces in P"1 that meet 16 JG' lines and go through 5 points. We have to compute: for 0 < / < a. For ji = 0 we get = f(-2c, JG' We may write c = CG + E]la(-2)16"1c,, where by Lemma 1 of the previous section, [f ; > 0 we liavc: CG= [ (-^MHM and C i= f JG' = f From Lemma 1, it follows that

16 c = 2 = 65536 x 1430 = 93716480. G 8 7 \ 9 The second ingredient for the computation is the numbers = f i:(iF-|*(B'i-In*( JG'

= / (-i .Since B' = i?,(OG.(B')) = -ci(OG.(-B') = -C,(OG.(I), we have: JG' From what we did before the example it follows that: .•'-(B') = -)'-c,(OG.(l)) = -Cl(OB'(l))- Now C?G'(~1) restricts to B' as the normal bundle of B in G, that we denote by Therefore we get: Since Sj = 0,j < 0, all c, vanish except c16 and we get:

Ci6 = / i'{c Af = -50. JB 2

So, c = cG + c16 = 93716480 - 50 = 93716430. We observe that if we want to compute the degre of the zero cycle [l'J"[p']6 with where », = rank(.VjiG) and JIJ(JVBG) denotes the j'h Segre class of A^BG. In order to o + 2b = d, b < n than the intersection above contains C C B'. (See §2.) It follows that compute the degree of cycles of this type we need to calculate the Segre classes of the the intersection considered is not proper and we have to consider a second blow-up in normal bundle of B in G. That is what we do next. order to get an enumeratively significant number. That is what we do in the next section.

10 5 Enumerative Calculus on the 2nd Blow-up where Sj(A/c.G') denotes the j'h Segre class Ac'G'. It follows that in order to calculate c(i,j) we need to compute the segre classes of A^oG' as well as ;'*(B')- That is what we Let G be the blow-up of G' along C. We have the diagram: do next. 2 Notation as in §2, let V = A £ 3 S2Q and C = P(V) xG(2in+1) C. G Theorem 3 The class of Arc-G' in the Grothendick ring-of C is given by:

2 At'G' = U + L ® S2M ®U + Q®C' where, D-Ov(-l) - V~>W-.O, is (Ae tautological sequence of O = P(V).

r Proof: We have C = P(V) C P(A BG) = B' c G'. So in tlie Grothendick ring of C, we have: where C is the exceptional divisor in G. Consider the diagram; Notation as in §4, let ! be the strict transform of V in G. By a reasoning analogous to Lemma 2, it follows that 1' contains C with multiplicity one. Recalling that 1' = irfl) - B', we get: C C P(ATBG) i = n"(ir(i) - B') - c = n'*(n*(i)) - n"(B') - c. The number of specializations of complete intersections of two quadrics that meet a lines and go through b points, with a + 2b = d, b < n can, therefore, be calculated as: B

Let B" = P(A'BG) |C . Then we have: where B = II'*(B') and p = (A e G such that p € base locus of A}. The zero cycle ' |C • above represents a proper intersection for all values of a and 6, In order to compute its degree we need to calculate: Therefore we get: r AbG' = tfc,B" + A B"B'

Since A/B"B' = p'(/tfcB), we get:

Letting a = (e,^)1—"^(^l)6, we have: |c. +0v(-l) = W(\) + Q ® t + 0v(- J 1 / C B'o= / CC'-'B'o = / - B'o= / where VT = L ® A2^£ ® ft ® 5 A< fits in the exact sequence: JG JG JG JG 2 0 _, V -* A^C'B" -* W -+ 0,

as in Theorem 2. Finally we get: Now, C = -C,0Q(1) and so i'l Let JIJ = )'oii.fc(P(A''BG)). We get: Ac'G' = C2 ® 5^M ©W + 2®£+i/, since A!£ = £®K..

r Denoting as before d = ront(52J ) and n + 1 = we conclude: (1)

12 Corollary X The class o/,Vc'G' in the Grothendick ring of C is given by:

JVOG' = U + dU ® C2 - (n + \)C ®U - (n + 1)C - C ® K

Proof: Over B the tautological sequence of G becomes: and the result follows. Applying the lemma above to equation (1), before Theorem 3 we gel:

0 — L © T -» S2T -> S2M — 0, C(«\J) = (2) and we get:

1 2 We now take advantage of the fact that C is a P^-bundle over C where i't = rank(S2Q) — A'oG = U + C Q (S2r- 1=1 „ I—l=[,?j—1. We recall the exact sequence. = U + d(U ®C2)- (n

Using the corollary above one ran compute the Chern classes and the Segre classes of Since rank(U) = j/2, it follows that cn+i(U) = 0. Therefore: A'c'G'. We haver =0 C(Ac'G') = C{.U)C{U®t2)iC(L®U)-in+l)C(L)n+lC{t®)Cy1.

Now, C(0V(-1)) = 1 -ui and therefore: Lot c.\(U) — ui,€,(£.) = /i),Ci(AL") = sioo,<%()£) = stoo- Then we have:

We get: 2 1 (1 + (2/»i - siooH + (h\ - swoAi + awo)* )" . =0 1 Note tliat tlif ('hern classes above satisfy the following relations: u" = 0, where i/j = +1 2 In equation (2) above we can, therefore, substitute t<^ for the expression above (n + .In - •[)/•>/ + 1 since (lim{C){) = Jim(P"() ) + dim(P"( - 1) ) + rfjm(52<2)() = n + w - involving the Chern classes of V and uj, i < v2- We are reduced to computing: + 1 1 .? \ — U /»r = 0i since 0-*£->7->Q-t0is the tautological sequence of /-n>}, for j, < v2. P"; ^foiJ1 = 0, .SJDO = 0 since 0—> K, —* T -^ M. —»0is the tautological sequence of f>'(2, H + 1). Using Matheniatica, we can compute the Chern classes of A/*cGr' and then We get: the Segre classes. As for j'(B') we have: Lemma 4

j'-(B') = (-l)V,, where Ul = c,(Cv(l)). This doesn't vanish only for it = 1/3 and so we get:

Proof: Consider the diagram:

C We have proved the following: Theorem 4 Fix a line I in P", Let \

14 \. when ii^'i, 1, the strict transform of V in G, parametrizes the set of specializations And so we get the formula: /i, = —^200 + 'ii^ioo-

of elliptic quartics incident to I and we have: On the other hand. V = £ 0 H 0 $7Q, (notation as in §2.) It follows that the Chern

classes of V can be computed in terms of Ai, .S]oo:*2oo and c(3. 17) can then be computed I16 = 52832040. using Schubert Calculus in G(3, 5). Let .s^ooi-Silo be a ba.sis for the Chow ring of G(3,5) 2 in codimension 2. Then we have s lm = s-joo + •'no (Pieri's formula) and we get: 5. when "=•/. 1 parametrizes the set of specializations of Del Pezzo Surfaces incident l = 0. to I and W( have: 26 j\ = 47867287590090 Using this all c(i,j) can be computed. In particular c(3,17) = —1206630.

Proof: We have already proved most of what is claimed in the theorem. Let us see how one gets the number in item 5 above. ACKNOWLEDGMENTS

The author wishes to express his gratitude to Israel Vainsencher who, 26 first of all, taught him the subject that eventually led to this work, and £ f also for the many discussions during the preparation of this paper. He thanks the Mathematics Department of UFPe for their hospitality during part of the period this research was being done. He would also like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for By Lemma 1 we have: hospitality at the International Centre, Trieste.

2r Let h{i) = [G. c,(A) '~'B". In §4, we showed how to compute numbers of this kind. With the help of Mathematics all the />(/) can be computed. 2

r(3. IT) = 257120/i/'slOO3 + 99640ft/3sl004 + 212}6hl'2sl006 + 2090ft/slOO*-

msl006chernV(\) where chcn>V[i) denotes the i"' Cheni class of V. This number is meant to be computed in C. Now, we note that, since C = P{K) —> G(2,n + 1) is a P'-bundle, we have the exact sequence: 0-.£V(-l) — £-. 2-*0.

Since rautiZ) = 1, it follows that c2Z = 0 and therefore:

15 16 References [l] Avritwr, I).: Complete Intersections of two Quadrics in P", in preparation. (2] Kleimau, S.L.: The Transcersalily of a General Translate, Compositio Math. 28 (197-1), 2S7-297. [3] Kleiman, S. fj.: Prohhm 15. Rigorous Foundation of Schubert's Enumtrative. Calcu- lus, Proc. Syinpos. Pure Math. 28, AMS, (1976), 445-482. [4] Kidman. S. L.: Chaslts's Evttineiutive Theory of Conies: A Historical Introduction, M. A. A. Studies in Mathematics 20 (1980), 117-138.

[•r>] Vaiiiscnrlirr, I.; Avritzor, D.: Cowpactifying the Space of Elliptic Quartic Curves, Complex ProjiT.tive Geometry, Lou. Math. Soc. LNS 179, G. EUingsrud, C.Peskine, (i. Sarchi^ro & S.A. Struminr e«ls., Cambridge University Press, (1992), 47-58. [6] VainspnrliPr, I.: Classes Caracteristicas em Geometria Algebrica, 15° Coloquio Hrasilciio clc Matematira, IMPA. (1985).

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