Numbers for reducible cubic scrolls

MATHEMATICAL SCIENCES

Numbers for reducible cubic scrolls

Israel Vainsencher^I; Fernando Xavier^II

^IICEX - Departamento de Matemática - UFMG, Cx. Postal 702 Belo Horizonte 30123-970 Belo Horizonte, MG, Brasil

^IIDepartamento de Matemática - UFPb, Cidade Universitária - 58051-900 João Pessoa, PB, Brasil

^{Correspondence}

ABSTRACT

We show how to compute the number of reducible cubic scrolls of codimension 2 in (math blackboard symbol P)ⁿ incident to the appropriate number of linear spaces.

Key words: cubic scrolls, enumerative geometry.

RESUMO

Mostramos como calcular o número de rolos cúbicos redutíveis de codimensão 2 em (math blackboard symbol P)ⁿ incidentes a espaços lineares apropriados.

Palavras-chave: rolos cúbicos, geometria enumerativa.

1 INTRODUCTION

A cubic rational normal scroll of codimension 2 in

ⁿ is the residual intersection of 2 quadric hypersurfaces containing a codimension 2 plane. It is also equal to the locus of rank 1 of a 2 x 3 matrix

(cf. (Harris 1992) where the L_i, M_j denote sufficiently general linear forms in the homogeneous coordinates x₀,...,x_n of ⁿ. For n = 3, this is a twisted cubic. Enumeration of such curves has attracted some attention in the last two decades, and is now but an example in Gromov-Witten theory, cf. (Fulton and Pandharipande 1997), (Kock and Vainsencher 1999). No such general technique is known yet for higher dimension. For the case of cubic scrolls, a procedure to determine some of these numbers was explained in (Vainsencher and Xavier 2002).

The purpose of this note is to show an application of Bott's residual formula for the calculation of the number of reducible (n-2)-dimensional cubic scrolls in ⁿ satisfying suitable incidence conditions to linear subspaces. The whole point is the simple observation that, just as in the case of twisted cubics, a general reducible cubic scroll of dimension (n-2)- is the union of an (n-2)-plane with an (n-2)-dimensional quadric meeting along an (n-3)-plane.

2 THE PARAMETER SPACE

As customary in enumerative questions, we take the lead from the previous observation and consider the incidence variety

is naturally obtained as a tower of fiber bundles. Start with the grassmannian,

of projective (n-3)-dimensional subspaces of ⁿ. It carries a tautological exact sequence

where rank = 3 and denotes the trivial bundle with fiber the dual space . The homogeneous coordinates x₀,...,x_n form a basis for the latter space. The fiber of over p = [n-3] Î is the 3-dimensional space of linear forms vanishing on p. The choice of a space = [n-2] containing a fixed p = [n-3] is tantamount to picking a 2-dimensional vector subspace of the fiber _p, i.e., a point in the dual space . Thus, the set of pairs p Ì is the total space of the ²-bundle, . Likewise, hyperplanes h through p form the total space of the ²-bundle, . The choice of a quadric k Ì h containing p yields the ^2n-2- bundle,

Here stands for the rank-(2n-1) vector bundle over () obtained as follows. Let be the kernel of . Since 3n = , this is the rank-3n bundle of quadrics vanishing at the varying p Î . Let = (-1) be the line bundle over () with fiber over h given by the scalar multiples of the linear form h. Put = /. Then fits into the diagram of natural vector bundle maps,

We may summarize the previous discussion as follows.

LEMMA 2.1. Notation as above, the parameter space is isomorphic to the fiber product,

and dim = 5n-4.

REMARK 2.2. Let

It should be noted that ® is not a flat family. It can be rendered flat by a single blowup along a suitable smooth subvariety of using techniques as in (Vainsencher and Xavier 2002), but this is not needed in the sequel.

3 CYCLE CLASSES

Now we define the divisors corresponding to the relevant incidence conditions.

Fix a line

₀ = ¹Ì ⁿ and a plane p₀ = ²Ì ⁿ. We define

We also need a bit more of notation. We write

for the tautological sequence over (), where rank =2.

PROPOSITION 3.1. The cycle classes of the divisors defined above can be expressed as follows in terms of Chern classes of the natural bundles:

PROOF. Let = áx₃,...,x_nñ Ì be the space of n-2 linear forms cutting p₀ = ²Ì ⁿ. Let = / denote the rank-3 quotient. Studying the diagram of vector bundles over ,

it can be seen that p Î meets p₀ if and only if the slant arrow s: ® is not injective at p. Hence D_p0,p is the divisor of zeros of s. It follows that D_p0,p = c₁() = c₁(). The proof of 2. is similar. The last formula is slightly trickier. It suffices to establish it over an open subset the complement of which contains no divisor. Thus, we may restrict to the locus of h transversal to ₀. Let = áx₃,...,x_nñ Ì be now the space of equations for ₀ = ¹Ì ⁿ. Let = / be the rank 2 quotient. The composition ® ®

² is injective over the transversal locus. The quotient map

= / + can be thought of as evaluation of linear forms at the point of intersecion of ₀ with a varying, transversal hyperplane. It yields an evaluation map for quadratic forms, S₂® S₂ and induces S₂(/) ® S₂. Composing with , we find (-1) ® S₂. The latter vanishes precisely over (along the transversal locus). Since c₁(S₂) = 2c₁() = -2c₁(), the formula follows .

4 NUMBERS

We shall abuse notation and keep writing

for the pullback of these divisors to . The intersection of general translates is transversal by standard Bertini-Kleiman-Sard arguments (applied to (2.2) ® ⁿ). Taking i₁+i₂+i₃ = 5n-4 = dim such translates, we may find the number of elements (p, , k) of such that the subspace p = [n-3] meets i₁

²'s, the subspace = [n-2] meets i₂

¹'s and the (n-2)-quadric k meets i₃

¹'s. This is done by evaluating N_i1,i2: =

For the specific task of enumerating the reducible cubic scrolls, we take translates of the divisors D_p0,p and : = . We are now asked to compute

for i+j = dim . This is a purely mechanical matter with the help of either Schubert calculus (e.q., as implemented in (Katz and Strømme 1992) or Bott's residue formula (cf. (Bott 1967), (Meurer 1996)). Here are the ingredients for the latter. First find the fixed points of a -action on , induced by an action x_i

t^w_ix_i on ⁿ. Choosing the integral weights w_i sufficiently general, it can be checked that all fixed points are isolated, starting at , then climbing up the tower

Pick p₁ = áx₀, x₁, x₂ñ Î = r[n-3n], among the fixed points in . There are just three fixed points on the fiber ² = (_p1). Choose one, say p_1,1 = áx₀ñ (a hyperplane through p₁). The fiber of () over p_1,1 is

with the obvious 2n-1 fixed points. Pick p_1,1,1 = áñ. By the same token, the fiber of ® () over p_1,1,1 is ² = (á₀, ₁, ₂ñ), with _i = áx₀, x₁, x₂ñ/áx_j, x_kñ. We obtain thus the fixed point p_1,1,1,1 = ₀ (among a total of .3.(2n-1)·3 possible ones). The contribution of this fixed point to Bott's formula is given by evaluating the indicated Chern classes in the equivariant cohomology ring at the point. Explicitly, c₁() evaluates to w_i, because the fiber _p1 = áx₃,...,x_nñ decomposes into eigenspaces with weights w_i, i = 3... n. The term c₁() evaluates to -w₁-w₂. Next, the fiber of (-1) is the line spanned by the equation of the hyperplane áx₀ñ; dualizing, we see that c₁( (1)) yields -w₀. The fiber of (1) is dual to the line áñ hence contributes with -2w₁. We also need the weights of the fiber of the tangent bundle of . Starting at T_p1, we write the eigenspace decomposition Hom(áx₃,...,x_n,ñ áx₃,...,x_nñ) = åx_i/x_j, with i Î {0,1,2}, j Î {3,...,n}. Add to this the (decomposition of the) tangent along the fiber of () ® , at áx₀ñ, namely, x₁/x₀+x₂/x₀. Continuing, get the tangent along the fiber of () ® (), to wit, (x₁x₂+···+x₁x_n++···+x₂x_n)/. At last, add the fiber of T( () /) at ,á₀ñ x₀/x₁+x₀/x₂. The top Chern class needed is the product of weights (w_i-w_j), i Î {0,1,2}, j Î {3,...,n}, times (w₁-w₀)(w₂-w₀), times (w₂-w₁)···(w_n-w₁)(2w₂-2w₁)···(w₂+w_n-2w₁), times (w₀-w₁)(w₀-w₂); call this product w. The total contribution of the present fixed point is the fraction

Quite miraculously, adding up the contributions over all fixed points, we get an integer.

The table below compiles some examples for 3 < n < 6.

Thus we find in

⁴ the numbers N₃ = 7484880 of reducible cubic scrolls È k « (p, , k) such that the line p = ¹Ì ⁴ meets 3 general given ²Ì ⁴ and p Í Ç k and the union È k meets dim -3 = 5·4-4-3 = 13 general lines. On the same row of the table we read N_5,2 = 1570, the number of configurations (p, , k) Î with the line p meeting i₁ = 5 general ²'s, the plane meeting i₂ = 2 general lines and the quadric surface k meeting i₃ = 16-i₁-i₂ = 9 other general lines. The entries have been computed employing Greuel et al. 2001. A script can be downloaded from Vainsencher 2004.

CLOSING REMARKS

As kindly pointed out by the referee, for n > 5 all rational normal cubic scrolls of codimension 2 in ⁿ are cones (cf. XXX 1957). For n = 4,5, the irreducible singular cubic scrolls not contained in a hyperplane are cones over a scroll in a hyperplane. The enumeration of such cones satisfying suitable incidence conditions can be pursued by means of a natural fibration. We hope to report on this elsewhere.

ACKNOWLEDGMENTS

We thank the referees for several suggestions and corrections. This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

Manuscript received on January 19, 2004; accepted for publication on July 6, 2004; contributed by ISRAEL VAINSENCHER^*

• BOTT R. 1967. A residue Formula for Holomorphic Vector Fields. J Diff Geom 1: 311-330.
• FULTON W AND PANDHARIPANDE R. 1997. Notes on Stable Maps and Quantum Cohomology. Proc Symp Purę Math 62(II): 45-96 (alg-geom/9608011).
• GREUEL G-M, PFISTER G AND SCHÖNEMANN H. 2001. Singular 2.0. A Computer Álgebra System for Polynomial Computations. Centre for Computer Álgebra, University of Kaiserslautern. http://www.singular.uni-kl.de
• HARRIS J. 1992. Algebraic geometry. A first course, Texts in Matematics, 133. Springer-Verlag, New York.
• KATZ S AND STRŘMME SA. 1992. Schubert. A maple package for Intersection Theory. http://www.mi.uib.no/schubert/
• KOCK J AND VAINSENCHER I. 1999. Kontsevich's formula for rational plane curves, book, to appear, http://www.mat.ufmg.br/ israel/kontsevich.html,
• MEURER P. 1996. The number of rational quartics on Calabi-Yau hypersurfaces in weighted projective space P(2,14) Math Scandin 78: 63-83 (alg-geom/9409001).
• VAINSENCHER I AND XAVIER F. 2002. A compactification of the space of twisted cubics. Math Scand 91: 221-243.
• VAINSENCHER I. 2004. http://www.mat.ufmg.br/ israel/sing.html
• XXX (ANONYMOUS) 1957. Correspondence. Am J Math 79: 951-952.

*

Member Academia Brasileira de Ciências

Publication Dates

• Publication in this collection
  22 Nov 2004
• Date of issue
  Dec 2004

History

• Received
  19 Jan 2004
• Accepted
  06 June 2004