A COMPACTIFICATION OF THE SPACE OF TWISTED CUBICS

Transcription

1 MATH. SCAND. 91 (2002), A COMPACTIFICATION OF THE SPACE OF TWISTED CUBICS I. VAINSENCHER and F. XAVIER Abstract We give an eementary, expicit smooth compactification of a parameter space for the famiy of twisted cubics. The construction aso appies to the famiy of subschemes defined by determinanta nets of quadrics, e.g., cubic rued surfaces in P 4, Segre varieties in P 5. It is suitabe for appications of Bott s formua to a few enumerative probems. 1. Introduction A twisted cubic curve (twc) is the image in projective 3-space of the map [t,u] [t 3,t 2 u, tu 2,u 3 ], for a suitabe choice of homogeneous coordinates. According to Harris [7], This is everybody s first exampe of a concrete variety that is not a hypersurface, inear space, or finite set of points. Our aim is to give a simpe, expicit smooth compactification of a parameter space for the famiy of twisted cubics. Simpe means no need of GIT. Expicit is intended to be suitabe for appications of Bott s formua (as in Eingsrud and Strømme [3], Meurer [8]) to a few enumerative probems. Piene and Schessinger [9] have shown that the Hibert scheme component H of twisted cubics is a smooth projective variety of dimension tweve. Later, Eingsrud, Piene and Strømme [1] proved that the subvariety D of the Grassmann variety G(3, 10) of nets of quadrics of determinanta type (i.e., spanned by the 2 2 minors of a 2 3 matrix of inear forms) is a smooth variety. H is the bowup of D aong the subvariety of nets with a fixed component. Eingsrud and Strømme [2] have aso shown that D is a geometric quotient of the set of semistabe 2 3 matrix of inear forms. This description enabed them to compute the Chow rings of D and H. A major motivation was to give a mathematica treatment to the physicists prediction for the number of twisted cubics contained in certain Caabi-Yau manifods (cf. [3]). The first author is partiay supported by CNPQ, the second by CAPES. Received November 15, 1999.

2 222 i. vainsencher and f. xavier We offer an aternative approach that eads utimatey to compactification of the set of a twcs that miss a fixed point o P 3. The main idea is best expained in the picture beow. C o If the twisted cubic C misses the point o, then there is a unique ine o that is bisecant (possiby tangent) to C. Now the configuration C is a compete intersection of a penci of quadrics. We simpy revert the construction. For each ine P 3, take the Grassmann variety G(2, 7) of pencis of quadrics containing. This defines a Grassmann bunde X over the Grassmannian of ines. There is a Zariski open subset of X that parametrizes a famiy of twcs. This famiy of twcs does not extend to a fat famiy over X. The main resut is the construction of a sequence of three bowups X X X X aong smooth, expicit centers that yieds a fat famiy of twcs over X. The first bowup fixes the probem of assigning a we defined net of quadrics to any penci of quadrics as above. Precisey, we get a morphism X G(3, 10) that extends the rationa map X G(3, 10) given by the net of quadrics through the residua twc determined by the penci. The remaining two bowups are designed to resove the indeterminacy of the rationa map X G(10, 20) defined by the system of cubics through a possiby degenerate twc. This is done by studying a suitabe saturation ( 3) of the subsheaf of the free sheaf of cubic forms that is the image of the natura map (cf. (18)) given by mutipying by inear forms a quadrics in a net. The construction aso appies with obvious minor changes to P n, for any n 2. Precisey, we get a simiar description for a smooth compactification of the famiy of subschemes of P n defined by nets of quadrics of determinanta type. As an amusing appication, we may retrieve the number 15 of trianges in P 2 meeting 6 genera ines and Schubert s 80,160 twisted cubics in P 3 meeting 12 genera ines. We aso find 648,151,945 (resp. 7,265,560,058,820) rationa

3 a compactification of the space of twisted cubics 223 rued cubic surfaces in P 4 (resp. Segre varieties in P 5 ) meeting 18 (resp. 24) genera ines. We note that a variation of a script of P. Meurer [8] adapted by A. Meirees takes a few seconds in a PC to produce these ast numbers, whereas the computation of Gromov-Witten invariants impemented by J. Kock, using Kresch s FARSTA [6], took about 3 days for n = 4 and was too big even to get started for n = 5. It aso reproduces the number (cf. [3]) of twisted cubics in P 4 contained in a genera quintic. 2. Notation and preiminaries Let F denote the space of inear forms in the variabes x = x 1,x 2,x 3,x 4. Let G(2, F ) be the Grassmann variety of ines in P 3, with tautoogica sequence (1) 0 L F F 0 where rank L = 2. The fiber L for G(2, F ) is the vector space of inear forms that vanish on the ine. Set Q := Ker(S 2 F S 2 F ). The fiber Q for G(2, F ) is the vector space of quadratic forms that vanish on the ine. We ceary have rank Q = 7. We write (2) X = G(2, Q) G(2, F ) for the Grassmann bunde of pencis of quadrics containining a varying ine G(2, F ). Each eement x X may be thought of as a pair (π, ) such that π represents a penci of quadrics through the distinguished ine. The atter is the image of x in G(2, F ). Denote by (3) R > Q X the rank two tautoogica subbunde of Q X. We omit the easy proofs of the foowing. Lemma 2.1. (i) The orbit of the point x 0 = (π 0, 0 ) X given by the penci π 0 = x1 2,x 1x 2 with distinguished ine 0 = x 1,x 2 is the unique cosed orbit of X under the natura action of GL(F ). (ii) Let P(L ) be the projective bunde over G(2, F ) that parametrizes the pairs (h, ) such that h is a pane containing the ine. Let ι : G(2, F ) P(L ) X

4 224 i. vainsencher and f. xavier be defined by assigning to each (λ, (h, )) the penci of quadrics beonging to X = G(2, Q ) with fixed component h and varying part the penci of panes with axis λ. Let B denote the image of ι. Then we have the foowing. 1. ι is an equivariant embedding; 2. B is normay fat over G(2, F ). We may think of a point of B as a pair (λ, (h, )) cf. the picture beow. We aso introduce now another reevant subvariety Y 1 X. It is the ocus of pencis with a fixed pane and moving penci of panes with axis equa to the distinguished ine. (4) = B := Y 1 := h h 9 7 Remark. Roughy speaking, the rest of this work is designed to fi in the technicaities needed to compete these pictures in order to produce a honest (fat) degenerations of a twc. Lemma 2.2. We have a natura embedding of ˇP 3 G(2, F ) in X defined by mutipying by a varying pane the penci of panes through a distinguished ine. Denote by Y 1 the image. Then the scheme-theoretic intersection Y 1 B is isomorphic to the incidence subvariety ( h) of ˇP 3 G(2, F ). 3. Saturation Let A be an integra domain, P an A-modue and M P a submodue. We define the saturation of M in P by s M ={m P a A, a = 0, am M }. Thus s M is just the inverse image under the quotient map P P/M of the torsion submodue of P/M. The foowing facts are easy to check. 1. ss M = s M. 2. For any mutipicative system S A we have S 1 ( s M ) = s (S 1 M ). 3. For submodues M, M P, ifm f = M f for some nonzero f A then s M = s M. 4. If M P = A n is a ocay spit submodue then s M = M. One may define the saturation of a sheaf of modues over an integra scheme in view of 2 above.

5 a compactification of the space of twisted cubics 225 We register in the foowing emma the main steps used at each bowup in order to produce the saturation of certain subsheaves of S m F. It is inspired by Raynaud s strategy of fattening by bowing up [10]. Lemma 3.1. Let U be an integra affine variety with coordinate ring A and et [ ] Ir M = 0 f 1... f s be a trianguar (r +1) n matrix with entries in A, where I r denotes an identity bock of size r. Let M A n be the submodue spanned by the rows of M. Assume the idea J of (r +1)-minors is non zero. Let ρ : U G(r +1, C n ) be the rationa map defined by M. Let U U be the bowup of the scheme of zeros V = Z (J ) and et V denote the exceptiona divisor. Then we have the foowing. 1. The map ρ extends to a morphism ρ : U G(r + 1, C n ). 2. Suppose V is a compete intersection of codimension t in U and J = f 1,...,f t for some t s. Then U is the cosed subscheme of U P t 1 defined by f i x j = f j x i, 1 i, j t, where the x i denote homogeneous coordinates for P t Let U 0 = U (U C t 1 ) U P t 1 be the affine open subset given by x 0 = 1 and put V 0 = V U 0. Then there are reguar functions y 2,...,y s on U 0 such that f i = y i f 1 for 2 i s and the coordinate ring of V 0 is the poynomia ring (A/J )[y 2,...,y t ]. 4. The restriction of ρ to the open subset V 0 = V C t 1 of the exceptiona divisor is given by ρ (z, a) = M z + e r+1 + a 2 e r+2 + +a t e r+t + y t+1 (a)e r+t+1 + +y s (a)e r+s where a = (a 2,...,a t ) C t 1 and M z denotes the span of the first r rows of M at the cosed point z V, whereas the e i are the standard unit vectors in C n. 5. Put B := O(U 0 ). The saturation of the image of M B in Bn is a spit, free submodue with basis given by the first r rows of M together with the new generator, e r+1 + y 2 e r y s e n obtained by dividing the ast row of M by f 1, the oca equation of the exceptiona divisor.

6 226 i. vainsencher and f. xavier 4. The associated net Our first task wi be to resove the indeterminacies of the rationa map X G(3,S 2 F ) that assigns to a genera penci of quadrics through a ine the net of quadrics that cut the residua twisted cubic. For simpicity, we expain the procedure restricted to the fiber X 0 of X over the fixed ine 0 = x 1,x 2. The trick that has made the cacuations go through is the foowing trivia observation. Let π = f 11 x 1 + f 12 x 2,f 21 x 1 + f 22 x 2 be a genera penci of quadrics containing the ine 0 := x 1 = x 2 = 0. Here the f ij denote inear forms. Each point on the intersection of the two quadrics that ies off that ine must annihiate the determinant f 11 f 22 f 12 f 21. Aso reca that the residua twisted cubic is cut out by a determinanta net. This eads us to ook at the 2 2 minors of the matrix ( ) f11 f 21 x 2. f 12 f 22 x 1 Thus, consider the rationa map, ν : X G(3,S 2 F ) π π + f 11 f 22 f 12 f 21. A routine check shows that ν is indeed we defined, i.e., the assigned net is independent of the choice of generators of the penci. Moreover, the ocus where ν is a morphism contains the compement of the ocus of pencis with a fixed component. In fact, ν is aso defined at some points representing pencis the base ocus of which contain more than one ine, essentiay because each eement of X carries a distinguished ine. 5. Resoving the indeterminacies of ν We show next that B (cf. 2.1) is the ocus of indeterminacy of the rationa map ν. Proposition 5.1. Let X be the bowup of X aong B. Let E be the exceptiona divisor. Then we have the foowing. 1. the rationa map ν ifts to a morphism ν : X G(3,S 2 F );

7 a compactification of the space of twisted cubics the fiber of E over (λ, (h, )) B is the projective space of the quotient vector space Q λ /(h L λ ); 3. the restriction of ν to E is given by the rue where q P(Q λ /(h L λ )). (q, (λ, (h, ))) q +(h L λ ) Proof. Norma fatness of B over G(2, F ) ensures that the formation of the bowup commutes with base change. Thus we may restrict the verification to a fiber X 0, say over the distinguished ine 0 = x 1,x 2. Let Ẋ 0 be the standard neighborhood of x 2 1,x 1x 2 in X 0 with coordinate functions a 1, a 2, a 3, a 4, a 5, b 1, b 2, b 3, b 4, b 5 so that the two quadrics { q1 = x a 1 x 1 x 3 + a 2 x 1 x 4 + a 3 x a 4x 2 x 3 + a 5 x 2 x 4, q 2 = x 1 x 2 + b 1 x 1 x 3 + b 2 x 1 x 4 + b 3 x b 4x 2 x 3 + b 5 x 2 x 4 give a oca triviaization for the tautoogica rank two subbunde (3). Put f 11 = x 1 + a 1 x 3 + a 2 x 4, f 21 = x 2 + b 1 x 3 + b 2 x 4, f 12 = a 3 x 2 + a 4 x 3 + a 5 x 4, f 22 = b 3 x 2 + b 4 x 3 + b 5 x 4. We have q i = f ij x j. This enabes us to represent the rationa map ν by a 3 10 matrix. Indeed, the subspace spanned by q 1, q 2 and q 3 = f 11 f 22 f 21 f 12 can be written as the row space of the 3 10 matrix obtained by coecting coefficients of the quadratic monomias. The ordered basis we choose is formed by the seven monomias appearing in q 1, q 2, in that order, together with x 2 3, x 3 x 4, x 2 4.Wefind 1 0 a 1 a 2 a 3 a 4 a (5) M = 0 1 b 1 b 2 b 3 b 4 b b 3 b 4 b 5 a 3 α 1 α 2 α 3 α 4 α 5 where we have set for short α 1 = a 1 b 3 a 3 b 1 a 4, α 2 = a 2 b 3 a 3 b 2 a 5, α 3 = a 1 b 4 a 4 b 1, α 4 = a 2 b 4 a 4 b 2 + a 1 b 5 a 5 b 1, α 5 = a 2 b 5 a 5 b 2.

8 228 i. vainsencher and f. xavier Adding to the third row b 3 times the second row in the above matrix, we see that the idea of 3 3 minors of M is spanned by (6) b 4 b 3 b 1, b 5 b 3 b 2, a 3 + b 2 3, a 4 a 1 b 3, a 5 a 2 b 3. This is the idea in Ẋ 0 of the subscheme V where ν is not defined. The subgroup G 0 of Aut(P 3 ) fixing 0 acts on X 0. Since the map ν is G 0 -invariant, so is V. Since G 0 is irreducibe, it foows that each irreducibe component of V is aso invariant. Indeed, et V 1 V be an irreducibe component. We have V 1 G 0 V 1 ; since the atter is irreducibe, the incusion is in fact an equaity as asserted. Therefore any irreducibe component must contain the unique cosed orbit and must show up in the present neighborhood. Hence V is in fact smooth and irreducibe. Soving the reations (6) for b 4, b 5, a 3, a 4, a 5 and pugging into q 1, q 2,we find that the penci degenerates to a penci of quadrics of the form { q1 = (x 1 + b 3 x 2 )(x 1 b 3 x 2 + a 2 x 4 + a 1 x 3 ), q 2 = (x 1 + b 3 x 2 )(x 2 + b 1 x 3 + b 2 x 4 ). Notice the appearance of a fixed component, namey the pane given by x 1 + b 3 x 2. It contains the distinguished ine 0. Thus we see that our penci of quadrics is in fact given now by that fixed pane times the penci of panes with axis equa to the ine λ = Z x 1 b 3 x 2 + a 2 x 4 + a 1 x 3,x 2 + b 1 x 3 + b 2 x 4. Recaing the definition 2.1 of B, this shows that, set-theoreticay, V and B agree. Since V is smooth, it foows that V = B. By genera principes (cf. 3.1), we have that the bowup X is the cosure in X G(3,S 2 F ) of the graph of ν. It remains to describe the behaviour of ν on the exceptiona divisor. This wi be done in the seque. Proposition 5.2. Notation as above, for q P(Q λ / (h Lλ )) we have the foowing. (i) If the ine λ is transversa to the pane h then the net of quadrics q + (h L λ ) defines a degenerate twc of the form λ κ, where κ denotes the conic Z h, q.

9 a compactification of the space of twisted cubics 229 k h (ii) If λ h then there are two possibiities. (1) The net of quadrics q +(h L λ ) sti defines a degenerate twc, equa to the union of a non-panar degree two structure on λ with another ine in the pane h. It is projectivey equivaent to a net of the ist x1 2,x 1x 3,x 1 x 3 + x 2 x 4, x1 2,x 1x 2,x 1 x 3 + x2 2, x1 2,x 1x 2,x 2 x 3, x1 2,x 1x 2,x2 2. (2) The net of quadrics acquires a fixed component and is projectivey equivaent to x 2 1,x 1x 2,x 1 x 3. This occurs aong a subvariety Y 2 E isomorphic to the variety of fags (7) Y 2 = {(p,,λ,h) P 3 G(2, F ) 2 ˇP 3 p λ h }. Y 2 = p h Proof. For (i) we may take the ine λ = Z x 3,x 4 and the pane h := x 2 = 0. Now q = ax 3 + bx 4 x 2 since q x 2 x 3,x 4. Hence the quadric Z (q) cuts the pane h in a conic κ. It foows easiy that Z q,x 2 x 3,x 2 x 4 is the union of κ and λ. For (ii), we start with the ine λ = Z x 1,x 2 and the pane h := x 1 = 0. Let q = ax 1 + bx 2 be a representative of a nonzero cass in Q 0 / x1 x 1,x 2. We may assume that the inear form a is in x 3,x 4 and b is in x 2,x 3,x 4. Check the cases. Suppose a = 0. If b is (resp. is not) a mutipe of x 2 we get the ast (resp. third) of the ist. If a = 0 we may take a = x 3.Nowb can t be zero est we get a fag as in (2). If x 4 appears in b, we may take b = x 4 and retrieve the first net of the ist. Presenty the net is of the form x 2 1,x 1x 2,x 1 x 3 + x 2 (αx 2 + βx 3 ). Ifβ = 0 we get the second of the ist. If

10 230 i. vainsencher and f. xavier β = 0 = α, we get the third of the ist (changing x 2 βx 2 + x 1 ). Finay, if αβ = 0 the net is equivaent to the second of the ist. Lemma 5.3. Suppose in (2) above that the pane be given by a inear form h, the ine λ by an additiona equation h and the point p by these previous two together with h. Map the fag h h, h h, h,h to the pair ( h 2,hh, h 2,hh,hh ) in the fiber of E over. Then this map is an embedding of the fag variety onto Y 2. Notice that Y 2 is of codimension six. Denoting by Y 2 the subvariety of B where λ h hods, we see that Y 2 sits over Y 2 as the P 1 subbunde of E Y 2 defined by P((h F )/(h L λ )). 6. Tangent maps and norma bundes We describe in this section the norma bundes of the embeddings B X Y 1 P(L ) G(2, F ) G(2, Q) ˇP 3 G(2, F ). We consider B, X, Y 1 as schemes over G(2, F ) (cf. 2 for notation). Care must be taken with B as there are two maps p,p λ : B G(2, F ). We take p as the structure map B G(2, F ); it factors through P(L ). In view of the formua for the reative tangent bunde of a Grassmann bunde, T X/G(2, F ) = Hom(R, Q/R), we must compute the restrictions of the tautoogica rank two subbunde (3) over B and over Y 1.Wehave T B/G(2, F ) = Hom(O L ( 1), L /O L ( 1)) Hom(L, F /L ), R B = O L ( 1) p λ L, R Y1 = O F ( 1) L. Proposition 6.1. Notation as above, we have the formuae for the norma bundes, (8) N Y1 /X = ( O F ( 1) 2 L )ˇ S3 F (2)/( O F ( 1) S 2 F (2)),

11 a compactification of the space of twisted cubics 231 where S m F (2) stands for the vector bunde / G(2, F ) with fiber over equa to the subspace S m F (2) of S m F of forms of degree m contained in the square of the homogeneous idea of. (9) ( N B/X = p 2 λ Ľ p (O L (2) 2 ) L ) p λ Q/(O L ( 1) L ). Before starting the proof, we need a quick review of mutiinear agebra. Given vector bundes E,F, reca the standard isomorphism Eˇ F = Hom(E, F ) given by ě f (e ě(e)f ). In terms of (oca) basis (e i ) E, with dua basis (ě i ) Ě and basis (f j ) F, (ϕ i,j ) Hom(E, F ) such that ϕ ij (e i ) = f j and ϕ ij (e k ) = 0, i = k, the isomorphism maps ě i f j to ϕ ij. We aso have for rank E = m, the isomorphism In particuar, for rank E = 2 we get m 1 E = Hom(E, m E) v 2 v m (v 1 v 1 v m ). (10) E = Hom(E, 2 E) = Eˇ 2 E e (e e e). In terms of a pair of oca dua basis e 1,e 2 and ě 1, ě 2,wehave { } e1 0 e 1 ě 2 e 1 e 2, e 2 e 1 e 2 { } (11) e1 e 1 e 2 e 2 ě 1 e 1 e 2, e 2 0 whence a 1 e 1 + a 2 e 2 (a 1 ě 2 a 2 ě 1 ) e 1 e 2. We may now proceed to the proof of the proposition N B/X. The reative tangent map T B/G(2, F ) > T G(2, Q )/G(2, F ) B at a point (,h,λ) P(L ) G(2, F ) = B is given by (12) Hom( h,/ h ) Hom(λ, F /λ) Hom(h λ, Q / h λ) (θ 1,θ 2 ) θ

12 232 i. vainsencher and f. xavier where θ (h h ) := θ 1 (h)λ + θ 2 (h )h + h λ. The partia derivative corresponding to the first factor may be written in the form, (13) / h Hom(λ, Q /h λ) h (h h h + h λ). We compose the above map with the isomorphisms (14) Hom(λ, Q /(h λ)) = (λ)ˇ ( Q /(h λ) ) = 2 λˇ λ ( Q /(h λ) ). Now notice that the space λ F maps onto h ˇ 2 Q λ /(h λ). Consequenty, we get a natura surjection 2 ( T G(2, Q )/G(2, F ) (,h,λ) = h ˇ λˇ λ Q /(h λ) ) h 2 2 Q λ /(h λ). Indeed, first repace the factor by ˇ 2. Then use the surjections ˇ h ˇ and λ F Q λ. In terms of a pair of dua basis {x 1,x 2 },{ˇx 1, ˇx 2 } ˇthe map is given by the rue (11), λ F = λ ˇ 2 F h ˇ 2 ( Q λ /(h λ) ) (15) {}}{ a x 1 b ˇx 2 h x 1 x 2 (ab), a x 2 b ˇx 1 h x 1 x 2 (ab), the bar now indicating cass mod h λ. To see that the map above factors through the natura surjection λ F λ ( Q / (h λ) ) we must show that ˇx 2 h (ax 2 ) = ˇx 1 h (ax 1 )

13 a compactification of the space of twisted cubics 233 hods in h ˇ ( Q λ /(h λ) ). This is equivaent to the condition ˇx 2 (h)(ax 2 ) = ˇx 1 (h)(ax 1 ) which is in turn a trivia consequence of the identity (mutipying by a λ) h =ˇx 1 (h)x 1 +ˇx 2 (h)x 2. Next we check that the image of (13) goes to zero in 2 λˇ h ˇ 2 ( Qλ /(h λ) ). Pick x and a 1,a 2 λ. Nowx + h is sent to the map in Hom( 2 λ, λ ( Q / (h λ) ) ) defined by (16) a 1 a 2 a 2 (a 1 x) a 1 (a 2 x). To show (16) goes to zero, we use the rue (15). For this, write x = c 1 x 1 +c 2 x 2. Then we see that a 2 x a 1 goes to (c 1 ˇx 2 c 2 ˇx 1 ) h (a 1 a 2 ). Therefore a 2 x a 1 a 1 x a 2 goes to zero as desired. The image of the partia derivative, Hom(λ, F /λ) Hom(h λ, Q / (h λ)) = 2 (λ h )ˇ h λ ( Q / (h λ) ) goes to zero in 2 λˇ h 2 2 ( Q λ /(h λ) ) as we. Indeed, etting ϕ Hom(λ, F /λ) and a 1,a 2 λ, set α i = ϕ(a i ) mod λ. Then ϕ is mapped to the homomorphism h 2 2 λ h λ ( Q / (h λ) ) given by h 2 a 1 a 2 h a 1 hα 2 h a 2 hα 1. We check that a 1 h α 2 λ F goes to zero in h ˇ 2 ( Q λ /(h λ) ). Empoying (15), we see that a 1 ( ˇx 1 (h)x 1 +ˇx 2 (h)x 2 ) α }{{} 2 goes to ( ˇx 1 (h) ˇx 2 (h) ˇx 2 (h) ˇx 1 (h))(a 1 α 2 ) = 0. h Summarizing, we have shown the formua (9). The proof of (8) is simiar and wi be omitted. 6.3 (End of proof of 5.1). Let us show that the the restriction of ν to the fibers of E B are as described in 5.1. We must show that the norma directions to B, say at (λ, (h, )) correspond to eements q P(Q λ / (h Lλ ))

14 234 i. vainsencher and f. xavier / and moreover, the assigned net is as prescribed there. Note that P(Q λ (h L λ )) is naturay isomorphic to the projectivization of the corresponding fiber in (9). The idea is to cacuate suitabe one parameter famiies of pencis. Take (λ, (h, )) in the open orbit of B. We may assume λ = x 3,x 4, h = x 1, = x 1,x 2. Write q = x 3 α + x 4 β, with α, β F. Form the one-parameter famiy by considering the matrix, [ ] x1 β α, t C. tx 2 x 3 x 4 We get a one-parameter famiy of pencis, π t = x 1 x 3 tx 2 β,x 1 x 4 + tx 2 α. Note that each quadric in π t does contain the distinguished ine. One checks that for genera t we have π t B. Indeed, if π t B for a t, we must have x 1 x 3 tx 2 β = ac, x 1 x 4 + tx 2 α = bc for some poynomias a, b, c in the variabes x, t. We may write c = x 1 + t c and simiary a = x 3 + tā, b = x 4 + t b. We get acx 4 + bcx 3 = tx 2 q. Canceing t, we obtain c( āx 4 + bx 3 ) = x 2 q. Setting t = 0 we have that x 1 divides q. This is contrary to the choice of q. We have that ν maps π t to the net π t + q whenever π t B. Letting t 0 we see that the norma direction defined by π t is the image of q in G(3,S 2 F ). Since the fibers of E and P(Q λ /(h L λ )) are projective spaces of the same dimension, the desired equaity foows over the open orbit at first, thence everywhere. This competes the proof of 5.1 Coroary 6.4. There are precisey two cosed orbits in X. One is represented by the point (17) o 1 = ( x2 1,x 1x 2, x 2 1,x 1x 2,x 1 x 3 ) G(2, Q ) G(3,S 2 F ). The other is represented by o 2 = ( x2 1,x 1x 2, x 2 1,x 1x 2,x 2 2 ) G(2, Q ) G(3,S 2 F ). Proof. We may restrict the search to the fiber X 0 (acted on by the stabiizer of 0, of course). We may aso start by setting λ = 0 = x 1,x 2, h = x 1. The genera eement q P(Q λ /(h L λ )) may be written in the form q = c 1 x 1 x 3 + c 2 x 1 x 4 + c 3 x c 4x 2 x 3 + c 5 x 2 x 4, [c 1,...,c 5 ] P 4.

15 a compactification of the space of twisted cubics 235 If c 3 = 0 we may act with x 3 tx 3, x 4 tx 4 and get o 2 in the orbit s cosure. If c 4 = 0orc 5 = 0 we change coordinates, x i x i + x 2 (i = 3or4) and reduce to the previous case. Finay, if q = c 1 x 1 x 3 + c 2 x 1 x 4 we may take a coordinate change (aways fixing x 1 and x 1,x 2 ) such that q = x 1 x 3. This produces o 1. A simpe argument shows that the two orbits are cosed. Remark 6.5. The orbit o 2 is irreevant in the seque. Indeed, the net of quadrics appearing in the second component of o 2 represents a degenerate twc. The mutipication map studied beow is of maxima rank at o 2 and the rationa map from X to Hib is reguar there. For this reason, the succeeding bowup centers wi be away from this orbit. 7. The mutipication map It turns out that the projectivized norma bunde of Y 2 (see (7)) in X at a genera point parametrizes the inear system of pane cubics passing through the point of intersection of the two ines and singuar at the distinguished point. Look at the picture beow. h This wi be shown by considering the natura mutipication map, (18) A F µ S 3 F, where A denotes the puback of the rank three tautoogica bunde of G(3,S 2 F ) via ν. The generic rank of µ is ten. It drops rank to nine aong a subscheme Y containing Y 2 as one of its two components (see (23)). Bowing up Y 2 in X brings us coser to the desired fat famiy of twcs. One further bowup is sti necessary, essentiay due to the ocus where λ = hods (whence the point of intersection is no onger determined). Actuay, in order to ensure that the abovementioned point in the intersection λ becomes everywhere we defined, a different bowup strategy wi be pursued beow. The other component of Y is a subvariety Y 1 X. Its fiber Y 1 over a distinguished ine G(2, F ) is isomorphic to the incidence variety Y 1 = { (p,h) P 3 ˇP 3 p h P 3}.

16 236 i. vainsencher and f. xavier p h Ceary, Y 1 is of codimension seven in X. It wi be shown beow that Y 1 is isomorphic to ˇP 3 bownup aong the penci P(L ) of panes containing the distinguished ine. Thus, Y 1 is the strict transform of Y 1 (see (4)) in X. 8. Loca cacuations The construction of the fat famiy of twcs invoves the cacuation of Fitting ideas of suitabe modifications of the sheaf coker (µ) (cf. 18). For this we need to compute a oca matrix representation for the mutipication map. We pick appropriate coordinate charts for the fibers X 0 of X (as in the proof of 5.1) and X 0 of X over the ine 0 G(2, F ) given by x 1 = x 2 = Loca chart for X 0 Recaing (6), it foows that the bowup of Ẋ 0 aong Ḃ 0 = B Ẋ 0 is covered by five affine pieces, one for each generator of the idea of the bowup center Ḃ 0. Since fatness is an open condition, it suffices to restrict to an affine chart Ẋ 0 containing the point o 1 which represents a cosed orbit (cf. 17). Take the oca equation of the exceptiona idea to be given by (19) ɛ = b 4 b 3 b 1. This choice is guided by the bueprint (3.1). Observe that in the matrix representation for ν (after suitabe row and coumn operations), the entry ɛ appears in the coumn corresponding to the monomia x 1 x 3. Dividing the third row of that matrix by ɛ, we see that it wi correspond to a quadric of the form x 1 x 3 + terms vanishing at the origin of the coordinate neighborhood, cf. (21) beow. The coordinate functions may be chosen as a 1,a 2,b 1,b 2,b 3,b 4,c 2,c 3,c 4,c 5. where the c i are the ratios to ɛ of the remaining four generators of the idea of Ḃ 0. Precisey, the map Ẋ 0 Ẋ 0 is given by the incusion of affine coordinate rings C[Ẋ 0 ] = C[a 1,...,b 5 ] C[Ẋ 0 ] = C[a 1,a 2,b 1,...,c 5 ]

17 defined by a compactification of the space of twisted cubics 237 (20) a 3 = b 2 3 c 3ɛ, a 5 = a 2 b 3 c 5 ɛ, a 4 = a 1 b 3 c 4 ɛ, b 5 = b 3 b 2 + c 2 ɛ. Presenty a oca basis for A is formed by the three quadrics q 1 = x1 2 + a 1x 1 x 3 + a 2 x 1 x 4 (b3 2 + c 3ɛ 1 )x2 2 + (a 1b 3 c 4 ɛ 1 )x 2 x 3 + (a 2 b 3 c 5 ɛ 1 )x 2 x 4, (21) q 2 = x 1 x 2 + b 1 x 1 x 3 + b 2 x 1 x 4 + b 3 x2 2 + (b 1b 3 + ɛ 1 )x 2 x 3 + (b 3 b 2 + c 2 ɛ 1 )x 2 x 4, q 3 = x 1 x 3 + c 2 x 1 x 4 + c 3 x (c 4 b 3 + c 3 b 1 )x 2 x 3 + (c 5 c 2 b 3 + c 3 b 2 )x 2 x 4 + (b 1 c 4 + a 1 )x (b 1 c 5 + a 1 c 2 + b 2 c 4 + a 2 )x 4 x 3 + (a 2 c 2 + b 2 c 5 )x 2 4 where q 1,q 2 yied a oca basis for the rank two tautoogica subbunde R > Q (cf. 2) pued back to X 0. A oca representation of the mutipication map µ may now be computed as a matrix in the form (22) I 9 0 R 0 0 where I 9 denotes an identity bock of size 9 and R is a row matrix that spans the Fitting idea J of minors of µ. Wefind that J is equa to the sum of the two ideas J 0 = a 1 + 2b 3 b 1 ɛ,a 2 + 2b 3 b 2 c 2 ɛ,c 3,c 4 2b 3,c 5 2c 2 b 3 and ɛ b 1,b 2, with ɛ = b 4 b 3 b 1 as in (19). Put (23) J 1 = J 0 + b 1,b 2, J 2 = J 0 + ɛ. Hence, J = J 1 J 2 hods. So the ocus where µ drops rank is the union of the two smooth pieces Y 1, Y 2 given ocay by the respective ideas J 1, J 2.

18 238 i. vainsencher and f. xavier Description of Y 1 Here are, freshy picked from the generators of J 1, the seven reations that define Y 1 ocay, (24) a 1 = b 4,b 1 = 0, b 2 = 0, a 2 = b 4 c 2,c 3 = 0, c 4 = 2b 3,c 5 = 2c 2 b 3. Substituting in (21), it can be seen that the net of quadrics acquires a fixed component, namey the pane given by (25) x 1 + b 3 x 2 + b 4 x 3 + b 4 c 2 x 4. Note that, in genera, this pane does not contain the distinguished ine. The moving part of the net, namey the net of panes x 1 b 3 x 2,x 2,x 3 + c 2 x 4, defines the point p = [0:0: c 2 :1] P 3. It is the point of intersection of the fixed pane with the distinguished ine. Moreover, ooking at the coefficients of the equation (25) that defines the pane, we recognize Y 1, as the dua space ˇP 3 bown up aong the penci of panes through the distinguished ine 0 := x 1 = x 2 = 0. Equivaenty, Y 1, is the cosure of the graph of the rationa map ˇP 3 0 = P 1 produced by intersecting a moving pane with the distinguished ine. The intersection of Y 1, with the exceptiona divisor E is equa to the exceptiona divisor of the bowup Y 1, ˇP 3. It corresponds to a choices of a pane through the distinguished ine, together with a marked point on the ine. We summarize the above discussion as foows. Lemma 8.1. Let Y 1 P(F ) ˇP 3 consist of a (p,,h)such that p is a point in the intersection of the ine with the pane h. Then Y 1 maps isomorphicay onto the strict transform of Y 1 in X X G(3,S 2 F ) Description of Y 2 This is just the fag variety (5.3) introduced earier. Indeed, soving the equations defined by the generators of J 2,wefind a 1 = 2b 1 b 3,a 2 = 2b 2 b 3,b 4 = b 1 b 3,c 3 = 0, c 4 = 2b 3,c 5 = 2c 2 b 3. Pugging these reations in (21) we get the net of quadrics with fixed pane x 1 + b 3 x 2 and moving part x 1 b 3 (b 2 x 4 + b 1 x 3 ), x 2 + b 1 x 3 + b 2 x 4,x 3 + c 2 x 4 fitting the prescription 5.3. We note that Y 2 is contained in the exceptiona

19 a compactification of the space of twisted cubics 239 divisor E, as the oca equation (19) of the atter is contained in the idea of the former. The ine λ is given by the first two inear forms. We see that λ coincides with the distinguished ine x 1,x 2 if and ony if b 1 = b 2 = 0 hods. These are the oca equations for Y 1 Y 2 in Y 2. We may summarize this as foows. Lemma 8.2. The intersection of Y 2 and Y 1, viewed inside Y 2, is equa to the codimension two subvariety of Y 2 where the two ines,λ coincide. Moreover, the strict transform Y 2 in the bowup X of X aong Y 1 is isomorphic to the cosure of the graph of the rationa map defined by (, λ) λ. In other words, (26) Y 2 ={(p, o,,λ,h) (P3 ) 2 (G(2, 4)) 2 ˇP 3 p λ h, o λ}. p h Remark 8.3. As a subset of Y 1, the intersection with Y 2 is just the codimension one subvariety where the fixed pane swaows the distinguished ine a in keeping a marked point as a reminder of an intersection point. Though a resut of Hironaka ([5], p. 41) ensures a commutativity of bowups, we have chosen to bowup X aong Y 1 first and then bowup aong the strict transform Y 2 of Y 2 due to a nice geometrica reason. Indeed, Y 2 carries a ocay spit subbunde of S 3F of cubic forms in the varying pane h that vanish at the point o and are singuar at the point p. In fact, Y 2 embeds in a punctua reative Hibert scheme parametrizing zero dimensiona subschemes of degree four of the famiy of panes in P 3 through a distinguished ine. A point (p, o,,λ,h)in Y 2 produces the subscheme p2 + o of the pane h. We mean by this the zero dimensiona subscheme of degree four defined by squaring the idea of p in h and intersecting with the idea of the point o. When p = o hods, we sti get a we defined imiting subscheme isomorphic to y 2,xy,x 3. Here we have taken h as the x,y-pane, = λ as the axis y = 0 and the point o = p = (0, 0). More precisey, et s be an affine coordinate in the ine. The equation of λ may be written as y = t(x s). The choice of p on λ wi be provided by intersecting λ with x = u. Thus, s, t, u are oca coordinates for (Y 2 ),h. After homogenizing, we find (e.g., using mape) the equaity for s = u, x uz, y t(x sz) 2 x sz,y =[ utsz 2 +t(u+s)xz uyz+xy tx 2, st 2 (s 2u)z 2 + 2ut 2 xz + 2t(s u)yz t 2 x 2 + y 2,u 2 yz 2 2uxyz + x 2 y,

20 240 i. vainsencher and f. xavier su 2 z 3 u(u + 2s)xz 2 + (2u + s)x 2 z x 3 ]. It can be checked that for a s, t, u the homogenous idea in the right hand side imposes 4 independent conditions on quartics (as we as aready on cubics). Since Hib 4 P embeds in the grassmannian of codimension 4 subspaces 2 of S 4 x,y,z, it foows that (Y 2 ),h maps into that Hib. In fact, one checks that the map to that grassmannian is an embedding. Assume this picture for the moment and et s see how do the exceptiona divisor E Y 1 and the strict transform Y 2 fit together. Now Y 2 E is the ocus where λ = hods. It sits over Y 2 E = Y 2 Y 1 as the P1 -bunde defined by varying the point denoted by o on the ine λ. The fiber of the projectivized norma bunde E of Y 2 X at a point given by (p, o,,λ,h)is the inear system of cubics in the pane h containing the subscheme p 2 + o described above. This corresponds to the picture at the beginning of 7. Detais are given in the next sections Bowing up Y 1 X Let X X be the bowup of Y 1 X. There are two interesting charts for this bowup, namey, those given by choosing as oca equation for the exceptiona divisor either b 1 or b 2 among the seven generators given in (24). We now et ɛ = b 1 be the chosen one. (The cacuations for b 2 are simiar and wi be omitted.) The bowup map is given by the reations, (27) a 1 = (d 5 3b 3 )b 1 + b 4, a 2 = (d 6 b 3 (2d 7 + c 2 ))b 1 + c 2 b 4, b 2 = d 7 b 1, c 3 = d 2 b 1, c 4 = 2b 3 + d 3 b 1, c 5 = 2c 2 b 3 + d 4 b 1 where b 1,b 4,b 3,c 2,d 2,...,d 7 are oca coordinates. Pug the above reations into the oca equations for Y 2 (cf ). We find, b 4 b 1 b 3 + d 5 b 1, c 2 (b 4 b 3 b 1 ) + d 6 b 1, b 4 b 3 b 1, d 2 b 1, d 3 b 1, d 4 b 1 for the idea of the tota transform. The strict transform is given by the saturation with respect to b 1, the oca equation of the exceptiona divisor. Hence Y 2 is ocay given by the idea, (28) b 4 b 3 b 1,d 2,d 3,d 4,d 5,d 6. We compute next the saturation of the image sheaf of the mutipication map µ pued back to X. Let M be the submodue spanned by rows of the matrix M obtained from (22) by puback via (27). We are required to find the row matrices R with entries in the coordinate ring of the present chart, such that b 1 R ies in M.

21 a compactification of the space of twisted cubics 241 Now the tenth row of M is of the form b 1 R 0 (i.e., a entries in that row are divisibe by b 1 ). Hence s M contains R 0. The first nonzero entry of R 0 is d 2 ; it appears in the 11th. coumn. Hence s M = M + R 0 hods since they agree upon inverting d 2. This aso shows that we get a oca presentation of s M given by repacing the row b 1 R 0 in M by R 0. The Fitting idea of s M defined by the minors of the presentation is spanned by the entries of R 0. This idea is precisey (28). It defines the ast bowup center Y 2. Soving (28) for b 4,d 2,...,d 6 and substituting in (21) pued back to the present coordinate chart, we find that the net of quadrics parametrized by Y 2 is of the form, (x 1 + b 3 x 2 ) times the net of panes x 1 b 1 b 3 (x 3 + d 7 x 4 ), x 2 + b 1 (x 3 + d 7 x 4 ), x 3 + x 4 c 2. As in (8.1.2), the ine λ is given by the first two generators of the atter net. We see that λ intersects the distinguished ine 0 = x 1,x 2 at the point o = [0:0: d 7 : 1]. This point is we defined even when the ines coincide. Ceary λ = 0 if and ony if b 1 = 0. This is of course in agreement with (26): Y 2 is the cosure of the graph of the rationa map Y 2 P3 defined by λ 0. The varying inear part of the net defines the point p = [ b 3 b 1 (c 2 d 7 ) : b 1 (c 2 d 7 ) : c 2 : 1] Bowing up Y 2 X Let X X be the bowup of Y 2 X. The discussion above shows that the saturation of the puback of the image of µ is a rank 10, ocay spit subsheaf of S 3 F. Hence X maps to G(10,S 3 F ) and dominates the Hib of twcs embedded in that grassmannian. In the neighborhood where ɛ = d 3 is a oca generator for the (ast) exceptiona divisor E Y 2, the bowup is given by the reations b 4 = b 3 b 1 + e 2 ɛ, d 2 = e 3 ɛ, d 4 = e 4 ɛ, d 5 = e 5 ɛ, d 6 = e 6 ɛ. For the sake of competeness we aso ist the strict transforms of the previous two exceptiona divisors. They are given by ɛ = e 2 and ɛ = b 1. This shows that our compactification X is buit from a suitabe open subset of the Grassmann bunde X by adding three norma crossing divisors. We aso observe that the tenth cubic, i.e., the one corresponding to the ast nonzero row of the oca matrix presentation, is given by, e 3 x (1 + 2e 3b 1 )x 3 x (2e 3d 7 b 1 + e 4 )x 4 x (e 5 + 2b 1 + e 3 b 2 1 )x2 3 x 2 + (b e 2 + e 5 b 1 )x ((2d 7 + 2e 4 )b 1 + e 5 c 2 + e 6 + 2e 3 d 7 b 2 1 )x 4x 2 x 3 + (d 2 7 e 3b d 7e 4 b 1 + e 6 c 2 )x 2 4 x 2

22 242 i. vainsencher and f. xavier + ((2d 7 + e 4 )b (e 6 + e 5 c 2 + d 7 e 5 )b 1 + d 7 e 2 + 2c 2 e 2 )x 4 x (d 2 7 e 4b d 7e 6 c 2 b 1 + d 7 c 2 2 e 2)x ((d 7 c 2 e 5 + e 6 c 2 + d 7 e 6 )b 1 + 2d 7 c 2 e 2 + c 2 2 e 2 + (2d 7 e 4 + d 2 7 )b2 1 )x2 4 x 3. Restricting to E,wefind that it cuts the fixed pane x 1 = b 3 x 2 in a cubic passing through o = [0:0: d 7 : 1] and singuar at p = [ b 3 b 1 (c 2 d 7 ) : b 1 (c 2 d 7 ) : c 2 : 1]. 9. Enumerative appications We expain how to find the number of twcs meeting 12 genera ines. First, the set of pencis of quadrics containing a distinguished ine such that the residua twisted cubic meet the ine x 3 = x 4 = 0 form a divisor L X.An easy eimination shows that in the coordinate chart for X 0, its oca equation is a 3 = b 2 3. Its tota transform in X 0 is given by c 3 times the oca equation ɛ = b 4 b 3 b 1 of E. Hence the strict transform is given by c 3 = 0. We see from (27) that the tota transform in X is given by d 2 b 1 whence the strict transform is d 2. Finay, a oca equation of the strict transform in X is found to be given by e 3. It foows that the cass of the strict transform L X can be written (omitting pubacks) as [L ] = [L] [E ] [E ] [E ]. In order to cut to size the 2 bisecant ines, we restrict X over the Schubert cyce G o G(2, 4) of ines meeting a fixed point o. The cass [L]inX is easiy computed as 2c 1 R + c 1 L. The cass [G o ]ing(2, 4) is c 2 L. The number sought for is therefore given by [L ] 12 c 2 L, and may be cacuated expicity using Bott s formua as in [8]. The script adaptaded by A. Meirees is avaiabe at the homepage [11]. It aso does the job of producing the higher dimensiona exampes cited at the introduction. Finay, et us sketch a verification of the enumerative significance of the above intersection theoretic computation. Let H o denote the restriction of X over G o. Let p : H o H be the map to the Hibert scheme component of twcs. Let GL o GL(F ) denote the stabiizer of the point o. Ceary p is GL o -invariant. Let H o be any irreducibe divisor shrunk by p, i.e., such that codim p( ) 2. We have that is GL o -invariant and so is its image p( ). Let I H be the divisor of twcs incident to a fixed ine away from o. One checks that I contains none of the two cosed GL o -orbits of H. Hence I

23 a compactification of the space of twisted cubics 243 contains no p( ). Therefore p 1 I is irreducibe. Hence it is equa to the cosure L of a suitabe open subset of L (restricted over G o ). Therefore, we may write [L ] 12 c 2 L = [p 1 I] 12 = [I] 12. The atter is we known to be an enumerativey significant Gromov-Witten invariant (cf. [4]). REFERENCES 1. Eingsrud, G., Piene, R., Strømme, S., On the variety of nets of quadrics defining twisted cubics, Space curves (Rocca di Papa, 1985), Lecture Notes in Math (1987), Eingsrud, G., Strømme, S., On the Chow ring of a geometric quotient, Ann. of Math. (2) 130, no. 1 (1989), Eingsrud, G., Strømme, S., The number of twisted cubic curves on the genera quintic threefod, Math. Scand. 76 (1995), Futon, W. and Pandharipande, R., Notes on stabe maps and quantum cohomoogy, Agebraic Geometry, Santa Cruz 1995 (J. Koár, R. Lazarsfed and D. Morrison, eds.), Proc. Symp. Pure. Math. 62, II, 45 96, ag-geom/ Hironaka, H., Smoothing of agebraic cyces of sma dimensions, Amer. J. Math. 90 (1968), Kresch, A., FARSTA, an agorithm for Gromov-Witten invariants of Grassmannians, in P. Auffi s Mittag-Leffer notes, Harris, J., Agebraic Geometry A First Course, Springer-Verag New York Inc., Meurer, P., The number of rationa quartics on Caabi-Yau hypersurfaces in weighted projective space P(2, 1 4 ), Math. Scand. 78 (1996), 63 83, ag-geom/ Piene, R., Schesinger, M., On the Hibert scheme compactification of the space of twisted cubics, Amer. J. Math., 107, no. 4 (1985), Raynaud, M., Fat modues in agebraic geometry, Compositio Math. 24 (1972), israe/twc.htm ISRAEL VAINSENCHER UFPE-DEPARTAMENTO DE MATEMÁTICA CIDADE UNIVERSITÁRIA RECIFE PE BRASIL E-mai: israe@dmat.ufpe.br FERNANDO XAVIER UFPB-DEPARTAMENTO DE MATEMÁTICA CAMPUS I-CIDADE UNIVERSITÁRIA JOÃO PESSOA PB BRASIL E-mai: fxs@dmat.ufpe.br